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Properties

Label 2-370-185.103-c1-0-10
Degree $2$
Conductor $370$
Sign $-0.166 - 0.985i$
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 + (0.304 + 1.13i)3-s + (−0.499 + 0.866i)4-s + (2.21 + 0.336i)5-s + (−0.832 + 0.832i)6-s + (0.993 + 3.70i)7-s − 0.999·8-s + (1.39 − 0.806i)9-s + (0.814 + 2.08i)10-s − 5.46i·11-s + (−1.13 − 0.304i)12-s + (0.988 − 1.71i)13-s + (−2.71 + 2.71i)14-s + (0.291 + 2.61i)15-s + (−0.5 − 0.866i)16-s + (−6.08 + 3.51i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.175 + 0.656i)3-s + (−0.249 + 0.433i)4-s + (0.988 + 0.150i)5-s + (−0.339 + 0.339i)6-s + (0.375 + 1.40i)7-s − 0.353·8-s + (0.465 − 0.268i)9-s + (0.257 + 0.658i)10-s − 1.64i·11-s + (−0.328 − 0.0879i)12-s + (0.274 − 0.475i)13-s + (−0.725 + 0.725i)14-s + (0.0751 + 0.675i)15-s + (−0.125 − 0.216i)16-s + (−1.47 + 0.852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.166 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.166 - 0.985i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.166 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27565 + 1.50980i\)
\(L(\frac12)\) \(\approx\) \(1.27565 + 1.50980i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 - 0.336i)T \)
37 \( 1 + (-4.05 + 4.53i)T \)
good3 \( 1 + (-0.304 - 1.13i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.993 - 3.70i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 5.46iT - 11T^{2} \)
13 \( 1 + (-0.988 + 1.71i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.08 - 3.51i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.64 - 0.440i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 6.08T + 23T^{2} \)
29 \( 1 + (-4.74 + 4.74i)T - 29iT^{2} \)
31 \( 1 + (0.415 + 0.415i)T + 31iT^{2} \)
41 \( 1 + (0.895 + 0.517i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 8.46T + 43T^{2} \)
47 \( 1 + (4.67 - 4.67i)T - 47iT^{2} \)
53 \( 1 + (-2.67 + 9.99i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.818 - 3.05i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.58 + 0.692i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-9.46 + 2.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.59 + 4.48i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.88 - 2.88i)T - 73iT^{2} \)
79 \( 1 + (-5.43 + 1.45i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-2.80 + 10.4i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (8.91 + 2.38i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 3.18iT - 97T^{2} \)
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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

−11.61950593723437110813025394604, −10.67551773077037243039768938511, −9.668105463406352193597882663507, −8.683725477987912602239353485783, −8.317998089156411717704974658368, −6.34250533649582252868186085801, −6.02522370896386044444442448128, −4.93402810472766241980240394832, −3.63616240901118086025894279045, −2.28700414487956240838492150275, 1.46836172998241924559133132475, 2.28667940533709440360008189962, 4.31599412291616453435211796384, 4.81215892617135270980780670229, 6.65878556130951594501181524150, 7.04137664515932351831592817827, 8.371558357562571097840001750394, 9.699702607177436460942217004922, 10.17905473334670925037724947552, 11.09743423821875084620701013142

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