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Properties

Label 2-370-185.23-c1-0-1
Degree $2$
Conductor $370$
Sign $0.0917 - 0.995i$
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  + (−0.866 − 0.5i)2-s + (−2.52 − 0.676i)3-s + (0.499 + 0.866i)4-s + (0.128 − 2.23i)5-s + (1.84 + 1.84i)6-s + (−3.97 − 1.06i)7-s − 0.999i·8-s + (3.32 + 1.92i)9-s + (−1.22 + 1.86i)10-s − 1.28i·11-s + (−0.676 − 2.52i)12-s + (0.274 − 0.158i)13-s + (2.90 + 2.90i)14-s + (−1.83 + 5.55i)15-s + (−0.5 + 0.866i)16-s + (0.512 − 0.887i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−1.45 − 0.390i)3-s + (0.249 + 0.433i)4-s + (0.0573 − 0.998i)5-s + (0.754 + 0.754i)6-s + (−1.50 − 0.402i)7-s − 0.353i·8-s + (1.10 + 0.640i)9-s + (−0.388 + 0.591i)10-s − 0.386i·11-s + (−0.195 − 0.729i)12-s + (0.0759 − 0.0438i)13-s + (0.777 + 0.777i)14-s + (−0.473 + 1.43i)15-s + (−0.125 + 0.216i)16-s + (0.124 − 0.215i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.0917 - 0.995i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.0917 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0319018 + 0.0290984i\)
\(L(\frac12)\) \(\approx\) \(0.0319018 + 0.0290984i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.128 + 2.23i)T \)
37 \( 1 + (-5.18 + 3.18i)T \)
good3 \( 1 + (2.52 + 0.676i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (3.97 + 1.06i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + 1.28iT - 11T^{2} \)
13 \( 1 + (-0.274 + 0.158i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.512 + 0.887i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.85 + 0.764i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 9.01iT - 23T^{2} \)
29 \( 1 + (-5.62 - 5.62i)T + 29iT^{2} \)
31 \( 1 + (-0.661 + 0.661i)T - 31iT^{2} \)
41 \( 1 + (9.70 - 5.60i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 - 5.24iT - 43T^{2} \)
47 \( 1 + (-2.65 + 2.65i)T - 47iT^{2} \)
53 \( 1 + (6.83 - 1.83i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.26 + 8.44i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (7.25 + 1.94i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.374 - 1.39i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.61 + 6.26i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.58 + 7.58i)T - 73iT^{2} \)
79 \( 1 + (4.64 + 1.24i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-1.52 + 0.409i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.06 - 0.284i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + 11.2T + 97T^{2} \)
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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

−11.62724609439900218409581614866, −10.75525741813502382570863648148, −9.834601962258815567692271624243, −9.122402542903805711215028275115, −7.83438665417915946301532105273, −6.70274714175738160964531896655, −6.01662660440115616313715544668, −4.87978679048953770881014672122, −3.39236880655985514180860602648, −1.21695291323168398239929501829, 0.04718837911470997367879313189, 2.69216344072715870478814851133, 4.32576584514363936877573257139, 5.80310559630035729218406887407, 6.42342050500424755797093676213, 6.90368036921809817778647453808, 8.456471085622796177993915286686, 9.781297357132472581453083589977, 10.23713526648891129384692036776, 10.87811793935059743455260805760

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