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Properties

Label 2-370-185.97-c1-0-8
Degree $2$
Conductor $370$
Sign $0.787 - 0.616i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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  + (0.5 − 0.866i)2-s + (−0.453 + 1.69i)3-s + (−0.499 − 0.866i)4-s + (1.94 − 1.10i)5-s + (1.24 + 1.24i)6-s + (−1.25 + 4.67i)7-s − 0.999·8-s + (−0.0652 − 0.0376i)9-s + (0.0112 − 2.23i)10-s + 1.80i·11-s + (1.69 − 0.453i)12-s + (−1.30 − 2.26i)13-s + (3.42 + 3.42i)14-s + (0.995 + 3.79i)15-s + (−0.5 + 0.866i)16-s + (3.01 + 1.73i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.262 + 0.977i)3-s + (−0.249 − 0.433i)4-s + (0.868 − 0.495i)5-s + (0.506 + 0.506i)6-s + (−0.473 + 1.76i)7-s − 0.353·8-s + (−0.0217 − 0.0125i)9-s + (0.00354 − 0.707i)10-s + 0.544i·11-s + (0.488 − 0.131i)12-s + (−0.362 − 0.628i)13-s + (0.914 + 0.914i)14-s + (0.257 + 0.979i)15-s + (−0.125 + 0.216i)16-s + (0.730 + 0.421i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.787 - 0.616i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.787 - 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49406 + 0.515102i\)
\(L(\frac12)\) \(\approx\) \(1.49406 + 0.515102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-1.94 + 1.10i)T \)
37 \( 1 + (-2.23 + 5.65i)T \)
good3 \( 1 + (0.453 - 1.69i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.25 - 4.67i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 - 1.80iT - 11T^{2} \)
13 \( 1 + (1.30 + 2.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.01 - 1.73i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.17 - 1.65i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 8.37T + 23T^{2} \)
29 \( 1 + (-5.16 - 5.16i)T + 29iT^{2} \)
31 \( 1 + (-6.16 + 6.16i)T - 31iT^{2} \)
41 \( 1 + (2.24 - 1.29i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + 1.35T + 43T^{2} \)
47 \( 1 + (0.101 + 0.101i)T + 47iT^{2} \)
53 \( 1 + (0.698 + 2.60i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.77 + 10.3i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.11 + 1.37i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-9.52 - 2.55i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.81 + 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.21 - 3.21i)T + 73iT^{2} \)
79 \( 1 + (13.7 + 3.67i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (4.18 + 15.6i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-3.75 + 1.00i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 - 4.12iT - 97T^{2} \)
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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

−11.70508215758215586843117604964, −10.19055888167832082800025717888, −9.889680120466758355521223657542, −9.227008510486107268678492934115, −8.055294152284955017329349359469, −6.07136546878061167900882004340, −5.49836654078979229451680088066, −4.70871811071753969802889006772, −3.21860296967267196187593266689, −1.98909701786575726510845917730, 1.10303561728667746595960903056, 3.05506688129103808426738439434, 4.37923633800130678567933017320, 5.83643251536631232320508150517, 6.68825624486423337317311999361, 7.17359918571195377808924083163, 8.041879096658142552857252609180, 9.779107677721982922879341645259, 10.10343591882023852149456586837, 11.53520390725742458951746734269

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