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Properties

Label 2-175-175.108-c1-0-4
Degree $2$
Conductor $175$
Sign $0.0984 - 0.995i$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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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

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

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.0984 - 0.995i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.0984 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11966 + 1.01439i\)
\(L(\frac12)\) \(\approx\) \(1.11966 + 1.01439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line