Properties

Label 2-175-175.108-c1-0-13
Degree $2$
Conductor $175$
Sign $0.361 + 0.932i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0270 + 0.516i)2-s + (−1.27 − 1.03i)3-s + (1.72 − 0.181i)4-s + (−0.509 − 2.17i)5-s + (0.499 − 0.687i)6-s + (−2.33 − 1.24i)7-s + (0.302 + 1.90i)8-s + (−0.0630 − 0.296i)9-s + (1.11 − 0.322i)10-s + (1.94 + 0.412i)11-s + (−2.38 − 1.54i)12-s + (−2.78 − 5.46i)13-s + (0.578 − 1.24i)14-s + (−1.60 + 3.30i)15-s + (2.41 − 0.512i)16-s + (6.82 + 2.62i)17-s + ⋯
L(s)  = 1  + (0.0191 + 0.365i)2-s + (−0.736 − 0.596i)3-s + (0.861 − 0.0905i)4-s + (−0.227 − 0.973i)5-s + (0.203 − 0.280i)6-s + (−0.882 − 0.469i)7-s + (0.106 + 0.674i)8-s + (−0.0210 − 0.0988i)9-s + (0.351 − 0.101i)10-s + (0.585 + 0.124i)11-s + (−0.688 − 0.447i)12-s + (−0.772 − 1.51i)13-s + (0.154 − 0.331i)14-s + (−0.413 + 0.853i)15-s + (0.602 − 0.128i)16-s + (1.65 + 0.635i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.361 + 0.932i$
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: \((2,\ 175,\ (\ :1/2),\ 0.361 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.818669 - 0.560816i\)
\(L(\frac12)\) \(\approx\) \(0.818669 - 0.560816i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.509 + 2.17i)T \)
7 \( 1 + (2.33 + 1.24i)T \)
good2 \( 1 + (-0.0270 - 0.516i)T + (-1.98 + 0.209i)T^{2} \)
3 \( 1 + (1.27 + 1.03i)T + (0.623 + 2.93i)T^{2} \)
11 \( 1 + (-1.94 - 0.412i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (2.78 + 5.46i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-6.82 - 2.62i)T + (12.6 + 11.3i)T^{2} \)
19 \( 1 + (0.302 - 2.87i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (-4.32 + 0.226i)T + (22.8 - 2.40i)T^{2} \)
29 \( 1 + (2.61 + 3.60i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.199 - 0.447i)T + (-20.7 + 23.0i)T^{2} \)
37 \( 1 + (3.06 - 4.71i)T + (-15.0 - 33.8i)T^{2} \)
41 \( 1 + (-1.18 - 0.384i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.03 - 2.03i)T + 43iT^{2} \)
47 \( 1 + (-1.09 - 2.84i)T + (-34.9 + 31.4i)T^{2} \)
53 \( 1 + (-4.38 + 5.40i)T + (-11.0 - 51.8i)T^{2} \)
59 \( 1 + (-9.92 - 11.0i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (8.11 + 7.30i)T + (6.37 + 60.6i)T^{2} \)
67 \( 1 + (-0.00505 + 0.0131i)T + (-49.7 - 44.8i)T^{2} \)
71 \( 1 + (0.558 - 0.405i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-11.2 + 7.28i)T + (29.6 - 66.6i)T^{2} \)
79 \( 1 + (-4.18 + 9.40i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (6.44 - 1.02i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (1.52 - 1.69i)T + (-9.30 - 88.5i)T^{2} \)
97 \( 1 + (-4.31 - 0.683i)T + (92.2 + 29.9i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.36416862652112242513314393933, −11.94183757438961990513490250540, −10.58280639504854506924022294831, −9.645330218710896828966330361684, −8.037936778757126141299533787142, −7.25850863104145649767538722157, −6.10462232423042253355161296790, −5.39086041924798530414422158201, −3.40791739487613286868124083293, −1.05581242342103216654867651731, 2.53541390637865761505146931784, 3.76080201057333088255866410274, 5.48048506318083715507438527773, 6.68144714918192248297784202646, 7.30943766830375946966929651601, 9.326873835250235135723832265024, 10.11447873636936709548816104396, 11.08632606664375526075392119691, 11.71639259632231815237510024106, 12.40083275555979077017773987775

Graph of the $Z$-function along the critical line