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Properties

Label 2-370-185.8-c1-0-18
Degree $2$
Conductor $370$
Sign $-0.788 + 0.615i$
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 + (−0.765 − 2.85i)3-s + (0.499 − 0.866i)4-s + (2.10 − 0.751i)5-s + (−2.09 − 2.09i)6-s + (−0.920 − 3.43i)7-s − 0.999i·8-s + (−4.98 + 2.87i)9-s + (1.44 − 1.70i)10-s + 4.83i·11-s + (−2.85 − 0.765i)12-s + (3.57 + 2.06i)13-s + (−2.51 − 2.51i)14-s + (−3.76 − 5.44i)15-s + (−0.5 − 0.866i)16-s + (3.48 + 6.03i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.442 − 1.65i)3-s + (0.249 − 0.433i)4-s + (0.941 − 0.336i)5-s + (−0.854 − 0.854i)6-s + (−0.348 − 1.29i)7-s − 0.353i·8-s + (−1.66 + 0.959i)9-s + (0.457 − 0.538i)10-s + 1.45i·11-s + (−0.825 − 0.221i)12-s + (0.991 + 0.572i)13-s + (−0.672 − 0.672i)14-s + (−0.971 − 1.40i)15-s + (−0.125 − 0.216i)16-s + (0.845 + 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.788 + 0.615i$
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: \((2,\ 370,\ (\ :1/2),\ -0.788 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593530 - 1.72506i\)
\(L(\frac12)\) \(\approx\) \(0.593530 - 1.72506i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 + 0.751i)T \)
37 \( 1 + (3.80 + 4.74i)T \)
good3 \( 1 + (0.765 + 2.85i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.920 + 3.43i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 4.83iT - 11T^{2} \)
13 \( 1 + (-3.57 - 2.06i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.48 - 6.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.30 + 4.86i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 1.47iT - 23T^{2} \)
29 \( 1 + (2.37 + 2.37i)T + 29iT^{2} \)
31 \( 1 + (2.94 - 2.94i)T - 31iT^{2} \)
41 \( 1 + (-1.84 - 1.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 0.602iT - 43T^{2} \)
47 \( 1 + (2.33 - 2.33i)T - 47iT^{2} \)
53 \( 1 + (-0.651 + 2.42i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.34 + 0.628i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.69 - 6.31i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-8.93 + 2.39i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-5.05 + 8.75i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.08 + 9.08i)T - 73iT^{2} \)
79 \( 1 + (-3.81 - 14.2i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-0.985 + 3.67i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (2.51 - 9.39i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−11.13502775576033663236058918347, −10.42092829643416969017767119443, −9.327679102608971268736306274759, −7.893081366737605768174969396837, −6.86822549190301812912144187460, −6.42263997352976296918562582306, −5.30649742789441783191507983645, −3.93902990012615312952132765417, −2.06043857745506889910521755065, −1.22789868162629254617862376201, 2.91599412954172665587942566527, 3.61727591713861369100154190083, 5.43336792908798295379962651226, 5.50666557687624731501860044018, 6.37814193634157440757872112580, 8.345449284173630527379532219998, 9.140239571423139242254985154577, 9.942526005968185908007231000781, 10.88794422852072864334394835360, 11.54153144131401361880182544190

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