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Properties

Label 2-370-185.82-c1-0-16
Degree $2$
Conductor $370$
Sign $0.151 + 0.988i$
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 + (2.99 − 0.803i)3-s + (−0.499 + 0.866i)4-s + (−1.15 − 1.91i)5-s + (−2.19 − 2.19i)6-s + (3.30 − 0.886i)7-s + 0.999·8-s + (5.75 − 3.32i)9-s + (−1.08 + 1.95i)10-s + 4.03i·11-s + (−0.803 + 2.99i)12-s + (−1.90 + 3.29i)13-s + (−2.42 − 2.42i)14-s + (−4.99 − 4.82i)15-s + (−0.5 − 0.866i)16-s + (−2.47 + 1.42i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (1.73 − 0.464i)3-s + (−0.249 + 0.433i)4-s + (−0.515 − 0.856i)5-s + (−0.896 − 0.896i)6-s + (1.25 − 0.335i)7-s + 0.353·8-s + (1.91 − 1.10i)9-s + (−0.342 + 0.618i)10-s + 1.21i·11-s + (−0.232 + 0.865i)12-s + (−0.527 + 0.914i)13-s + (−0.647 − 0.647i)14-s + (−1.29 − 1.24i)15-s + (−0.125 − 0.216i)16-s + (−0.600 + 0.346i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.151 + 0.988i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.151 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45371 - 1.24751i\)
\(L(\frac12)\) \(\approx\) \(1.45371 - 1.24751i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.15 + 1.91i)T \)
37 \( 1 + (-2.54 - 5.52i)T \)
good3 \( 1 + (-2.99 + 0.803i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-3.30 + 0.886i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 - 4.03iT - 11T^{2} \)
13 \( 1 + (1.90 - 3.29i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.47 - 1.42i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.23 + 4.61i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.69T + 23T^{2} \)
29 \( 1 + (2.30 + 2.30i)T + 29iT^{2} \)
31 \( 1 + (-1.95 + 1.95i)T - 31iT^{2} \)
41 \( 1 + (-4.19 - 2.42i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.10T + 43T^{2} \)
47 \( 1 + (5.47 + 5.47i)T + 47iT^{2} \)
53 \( 1 + (6.82 + 1.82i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.59 - 1.23i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.89 - 10.8i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.41 - 5.28i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.05 - 8.75i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.11 + 7.11i)T + 73iT^{2} \)
79 \( 1 + (2.88 + 10.7i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-2.76 - 0.739i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (3.14 - 11.7i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 - 10.0iT - 97T^{2} \)
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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

−11.37235839355424489154564326584, −9.924551003888456721343369588417, −9.209777252843165241379897927557, −8.421930107756002545661240056547, −7.78673083257901996472698978969, −7.07093166991569334450746195667, −4.46933952358407045033000730743, −4.23467006031677594109168411933, −2.37239615356342471780728398166, −1.58815060084271872489174256408, 2.19653038485025742509663271360, 3.38033248381223332786897627913, 4.46884345599656663618596867267, 5.88782678150151591807356641976, 7.45252148542844342634667682382, 8.039463125656311662219494123888, 8.478280537389265854083307633516, 9.547294682563532074853542581987, 10.52855572479519736942007916482, 11.23365031326221409416752141865

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