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Properties

Label 2-370-185.8-c1-0-1
Degree $2$
Conductor $370$
Sign $-0.998 + 0.0568i$
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 + (0.803 + 2.99i)3-s + (0.499 − 0.866i)4-s + (−0.0397 + 2.23i)5-s + (−2.19 − 2.19i)6-s + (−0.886 − 3.30i)7-s + 0.999i·8-s + (−5.75 + 3.32i)9-s + (−1.08 − 1.95i)10-s + 4.03i·11-s + (2.99 + 0.803i)12-s + (−3.29 − 1.90i)13-s + (2.42 + 2.42i)14-s + (−6.73 + 1.67i)15-s + (−0.5 − 0.866i)16-s + (1.42 + 2.47i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.464 + 1.73i)3-s + (0.249 − 0.433i)4-s + (−0.0177 + 0.999i)5-s + (−0.896 − 0.896i)6-s + (−0.335 − 1.25i)7-s + 0.353i·8-s + (−1.91 + 1.10i)9-s + (−0.342 − 0.618i)10-s + 1.21i·11-s + (0.865 + 0.232i)12-s + (−0.914 − 0.527i)13-s + (0.647 + 0.647i)14-s + (−1.73 + 0.433i)15-s + (−0.125 − 0.216i)16-s + (0.346 + 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.998 + 0.0568i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.998 + 0.0568i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0256792 - 0.903373i\)
\(L(\frac12)\) \(\approx\) \(0.0256792 - 0.903373i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0397 - 2.23i)T \)
37 \( 1 + (-5.52 + 2.54i)T \)
good3 \( 1 + (-0.803 - 2.99i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.886 + 3.30i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 4.03iT - 11T^{2} \)
13 \( 1 + (3.29 + 1.90i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.42 - 2.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.23 - 4.61i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.69iT - 23T^{2} \)
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 \( 1 - 8.10iT - 43T^{2} \)
47 \( 1 + (5.47 - 5.47i)T - 47iT^{2} \)
53 \( 1 + (-1.82 + 6.82i)T + (-45.8 - 26.5i)T^{2} \)
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 \( 1 + (-5.28 + 1.41i)T + (58.0 - 33.5i)T^{2} \)
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 \( 1 + (0.739 - 2.76i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-3.14 + 11.7i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−11.29298713227088920845123265486, −10.29556101280719951970777790102, −10.16072022526853185721168758741, −9.558822955344094687592774331921, −8.098355937011294083817931604201, −7.40892336346333059870304430239, −6.21264744963204654907233773929, −4.75899082066756781044001734818, −3.87369325150179591502976041261, −2.68513261336204061322058028759, 0.69180518356468943798008916151, 2.12381037210825388510761882544, 3.07315965799864838751772150160, 5.30464575393977941898669869359, 6.30971889451603803112481111465, 7.41117146194210173423161120213, 8.234898829064699046129754235565, 8.992697390517093035468174701529, 9.476916152826930797540415290879, 11.46189544736164038972876425959

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