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Properties

Label 2-175-175.108-c1-0-6
Degree $2$
Conductor $175$
Sign $0.788 - 0.615i$
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  + (0.0712 + 1.35i)2-s + (−0.508 − 0.412i)3-s + (0.146 − 0.0153i)4-s + (2.05 − 0.886i)5-s + (0.524 − 0.721i)6-s + (1.38 − 2.25i)7-s + (0.457 + 2.88i)8-s + (−0.534 − 2.51i)9-s + (1.35 + 2.72i)10-s + (−3.52 − 0.749i)11-s + (−0.0806 − 0.0523i)12-s + (3.11 + 6.10i)13-s + (3.16 + 1.72i)14-s + (−1.41 − 0.395i)15-s + (−3.60 + 0.766i)16-s + (−0.931 − 0.357i)17-s + ⋯
L(s)  = 1  + (0.0503 + 0.961i)2-s + (−0.293 − 0.237i)3-s + (0.0730 − 0.00767i)4-s + (0.918 − 0.396i)5-s + (0.213 − 0.294i)6-s + (0.523 − 0.851i)7-s + (0.161 + 1.02i)8-s + (−0.178 − 0.838i)9-s + (0.427 + 0.862i)10-s + (−1.06 − 0.225i)11-s + (−0.0232 − 0.0151i)12-s + (0.862 + 1.69i)13-s + (0.845 + 0.460i)14-s + (−0.364 − 0.101i)15-s + (−0.901 + 0.191i)16-s + (−0.225 − 0.0866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.788 - 0.615i$
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: \((2,\ 175,\ (\ :1/2),\ 0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28857 + 0.443301i\)
\(L(\frac12)\) \(\approx\) \(1.28857 + 0.443301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.05 + 0.886i)T \)
7 \( 1 + (-1.38 + 2.25i)T \)
good2 \( 1 + (-0.0712 - 1.35i)T + (-1.98 + 0.209i)T^{2} \)
3 \( 1 + (0.508 + 0.412i)T + (0.623 + 2.93i)T^{2} \)
11 \( 1 + (3.52 + 0.749i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-3.11 - 6.10i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.931 + 0.357i)T + (12.6 + 11.3i)T^{2} \)
19 \( 1 + (0.467 - 4.44i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (3.95 - 0.207i)T + (22.8 - 2.40i)T^{2} \)
29 \( 1 + (4.04 + 5.57i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.51 + 5.65i)T + (-20.7 + 23.0i)T^{2} \)
37 \( 1 + (1.65 - 2.54i)T + (-15.0 - 33.8i)T^{2} \)
41 \( 1 + (-5.72 - 1.85i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-3.08 - 3.08i)T + 43iT^{2} \)
47 \( 1 + (1.75 + 4.56i)T + (-34.9 + 31.4i)T^{2} \)
53 \( 1 + (5.55 - 6.86i)T + (-11.0 - 51.8i)T^{2} \)
59 \( 1 + (-1.85 - 2.05i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-2.72 - 2.44i)T + (6.37 + 60.6i)T^{2} \)
67 \( 1 + (-5.05 + 13.1i)T + (-49.7 - 44.8i)T^{2} \)
71 \( 1 + (7.11 - 5.17i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.33 - 2.81i)T + (29.6 - 66.6i)T^{2} \)
79 \( 1 + (0.406 - 0.913i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (13.6 - 2.16i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (-1.15 + 1.28i)T + (-9.30 - 88.5i)T^{2} \)
97 \( 1 + (-13.1 - 2.08i)T + (92.2 + 29.9i)T^{2} \)
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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

−13.12713524771302154679403616916, −11.73141298704437608708681498806, −10.95740246943546478786311227035, −9.700220265356557506633636486063, −8.516686086344693733656446627746, −7.49199616659181103536909014486, −6.32294205021792669600463757380, −5.75722185178302110862197025548, −4.27640866405497567371138621295, −1.84473769368120531680832502530, 2.05432938172655929680153286928, 3.02490650807016513382747001792, 5.10082766750482691346733564092, 5.89085995014606681182908791360, 7.48790423841381288142423600221, 8.751908641271241480440062280419, 10.12723825764002943452433950648, 10.75547845314360968957103987044, 11.24519581150436788806725788347, 12.76470835416798495731675467345

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