↑

Properties

Label 2-175-175.108-c1-0-4
Degree 22
Conductor 175175
Sign 0.0984−0.995i0.0984 - 0.995i
Analytic cond. 1.397381.39738
Root an. cond. 1.182101.18210
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0691 + 1.31i)2-s + (1.21 + 0.985i)3-s + (0.253 − 0.0266i)4-s + (−0.928 − 2.03i)5-s + (−1.21 + 1.67i)6-s + (0.569 + 2.58i)7-s + (0.465 + 2.94i)8-s + (−0.113 − 0.536i)9-s + (2.61 − 1.36i)10-s + (0.468 + 0.0995i)11-s + (0.335 + 0.217i)12-s + (0.0993 + 0.195i)13-s + (−3.36 + 0.930i)14-s + (0.874 − 3.39i)15-s + (−3.34 + 0.712i)16-s + (−4.48 − 1.72i)17-s + ⋯
L(s)  = 1  + (0.0488 + 0.932i)2-s + (0.702 + 0.568i)3-s + (0.126 − 0.0133i)4-s + (−0.415 − 0.909i)5-s + (−0.496 + 0.683i)6-s + (0.215 + 0.976i)7-s + (0.164 + 1.04i)8-s + (−0.0379 − 0.178i)9-s + (0.828 − 0.431i)10-s + (0.141 + 0.0300i)11-s + (0.0967 + 0.0628i)12-s + (0.0275 + 0.0540i)13-s + (−0.900 + 0.248i)14-s + (0.225 − 0.875i)15-s + (−0.837 + 0.178i)16-s + (−1.08 − 0.417i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.0984−0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0984 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((0.0984−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0984 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.0984−0.995i0.0984 - 0.995i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(108,⋅)\chi_{175} (108, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), 0.0984−0.995i)(2,\ 175,\ (\ :1/2),\ 0.0984 - 0.995i)

Particular Values

L(1)L(1) ≈\approx 1.11966+1.01439i1.11966 + 1.01439i
L(12)L(\frac12) ≈\approx 1.11966+1.01439i1.11966 + 1.01439i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(0.928+2.03i)T 1 + (0.928 + 2.03i)T
7 1+(−0.569−2.58i)T 1 + (-0.569 - 2.58i)T
good2 1+(−0.0691−1.31i)T+(−1.98+0.209i)T2 1 + (-0.0691 - 1.31i)T + (-1.98 + 0.209i)T^{2}
3 1+(−1.21−0.985i)T+(0.623+2.93i)T2 1 + (-1.21 - 0.985i)T + (0.623 + 2.93i)T^{2}
11 1+(−0.468−0.0995i)T+(10.0+4.47i)T2 1 + (-0.468 - 0.0995i)T + (10.0 + 4.47i)T^{2}
13 1+(−0.0993−0.195i)T+(−7.64+10.5i)T2 1 + (-0.0993 - 0.195i)T + (-7.64 + 10.5i)T^{2}
17 1+(4.48+1.72i)T+(12.6+11.3i)T2 1 + (4.48 + 1.72i)T + (12.6 + 11.3i)T^{2}
19 1+(−0.561+5.33i)T+(−18.5−3.95i)T2 1 + (-0.561 + 5.33i)T + (-18.5 - 3.95i)T^{2}
23 1+(−0.532+0.0278i)T+(22.8−2.40i)T2 1 + (-0.532 + 0.0278i)T + (22.8 - 2.40i)T^{2}
29 1+(2.22+3.06i)T+(−8.96+27.5i)T2 1 + (2.22 + 3.06i)T + (-8.96 + 27.5i)T^{2}
31 1+(−1.36−3.05i)T+(−20.7+23.0i)T2 1 + (-1.36 - 3.05i)T + (-20.7 + 23.0i)T^{2}
37 1+(−4.15+6.39i)T+(−15.0−33.8i)T2 1 + (-4.15 + 6.39i)T + (-15.0 - 33.8i)T^{2}
41 1+(2.67+0.870i)T+(33.1+24.0i)T2 1 + (2.67 + 0.870i)T + (33.1 + 24.0i)T^{2}
43 1+(−6.03−6.03i)T+43iT2 1 + (-6.03 - 6.03i)T + 43iT^{2}
47 1+(2.68+6.98i)T+(−34.9+31.4i)T2 1 + (2.68 + 6.98i)T + (-34.9 + 31.4i)T^{2}
53 1+(3.04−3.76i)T+(−11.0−51.8i)T2 1 + (3.04 - 3.76i)T + (-11.0 - 51.8i)T^{2}
59 1+(3.58+3.97i)T+(−6.16+58.6i)T2 1 + (3.58 + 3.97i)T + (-6.16 + 58.6i)T^{2}
61 1+(−6.61−5.95i)T+(6.37+60.6i)T2 1 + (-6.61 - 5.95i)T + (6.37 + 60.6i)T^{2}
67 1+(4.65−12.1i)T+(−49.7−44.8i)T2 1 + (4.65 - 12.1i)T + (-49.7 - 44.8i)T^{2}
71 1+(−3.80+2.76i)T+(21.9−67.5i)T2 1 + (-3.80 + 2.76i)T + (21.9 - 67.5i)T^{2}
73 1+(−11.0+7.14i)T+(29.6−66.6i)T2 1 + (-11.0 + 7.14i)T + (29.6 - 66.6i)T^{2}
79 1+(3.76−8.46i)T+(−52.8−58.7i)T2 1 + (3.76 - 8.46i)T + (-52.8 - 58.7i)T^{2}
83 1+(9.38−1.48i)T+(78.9−25.6i)T2 1 + (9.38 - 1.48i)T + (78.9 - 25.6i)T^{2}
89 1+(3.91−4.35i)T+(−9.30−88.5i)T2 1 + (3.91 - 4.35i)T + (-9.30 - 88.5i)T^{2}
97 1+(9.51+1.50i)T+(92.2+29.9i)T2 1 + (9.51 + 1.50i)T + (92.2 + 29.9i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.10663367596706519060569665315, −11.89434643035056567933623955410, −11.16700694202696773859811726255, −9.335185230087903529924113755131, −8.855552035105303425174305913514, −7.980349346196537121763136748857, −6.68507343978791628535992436788, −5.40845577285403415592009071294, −4.37458025625311157148517628346, −2.54702590458905120205814054017, 1.79583302057052891125957213455, 3.09598681877467456717576306959, 4.14913559345846828433043188005, 6.49363847169130085422182418978, 7.37572557896693658407792498779, 8.193897279592724403797058353147, 9.821900921234564119923926582949, 10.78224748017914022928254267914, 11.27065423303392751673733034120, 12.47841457238126297234743660253

Graph of the ZZ-function along the critical line