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Properties

Label 2-370-185.82-c1-0-0
Degree $2$
Conductor $370$
Sign $-0.974 + 0.226i$
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 + (0.234 − 0.0628i)3-s + (−0.499 + 0.866i)4-s + (−2.06 − 0.856i)5-s + (0.171 + 0.171i)6-s + (−3.50 + 0.939i)7-s − 0.999·8-s + (−2.54 + 1.47i)9-s + (−0.290 − 2.21i)10-s + 0.0505i·11-s + (−0.0628 + 0.234i)12-s + (0.743 − 1.28i)13-s + (−2.56 − 2.56i)14-s + (−0.538 − 0.0711i)15-s + (−0.5 − 0.866i)16-s + (−4.48 + 2.58i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.135 − 0.0362i)3-s + (−0.249 + 0.433i)4-s + (−0.923 − 0.383i)5-s + (0.0701 + 0.0701i)6-s + (−1.32 + 0.354i)7-s − 0.353·8-s + (−0.848 + 0.490i)9-s + (−0.0919 − 0.701i)10-s + 0.0152i·11-s + (−0.0181 + 0.0677i)12-s + (0.206 − 0.357i)13-s + (−0.685 − 0.685i)14-s + (−0.139 − 0.0183i)15-s + (−0.125 − 0.216i)16-s + (−1.08 + 0.627i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.974 + 0.226i$
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: \((2,\ 370,\ (\ :1/2),\ -0.974 + 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0403811 - 0.352200i\)
\(L(\frac12)\) \(\approx\) \(0.0403811 - 0.352200i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.06 + 0.856i)T \)
37 \( 1 + (-6.03 - 0.784i)T \)
good3 \( 1 + (-0.234 + 0.0628i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (3.50 - 0.939i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 - 0.0505iT - 11T^{2} \)
13 \( 1 + (-0.743 + 1.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.48 - 2.58i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0622 - 0.232i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 0.131T + 23T^{2} \)
29 \( 1 + (-2.88 - 2.88i)T + 29iT^{2} \)
31 \( 1 + (-0.644 + 0.644i)T - 31iT^{2} \)
41 \( 1 + (2.11 + 1.22i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 8.89T + 43T^{2} \)
47 \( 1 + (6.59 + 6.59i)T + 47iT^{2} \)
53 \( 1 + (2.46 + 0.660i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-9.63 - 2.58i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.655 - 2.44i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.27 - 8.48i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.62 - 9.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.387 + 0.387i)T + 73iT^{2} \)
79 \( 1 + (1.65 + 6.16i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (8.90 + 2.38i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (4.37 - 16.3i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 - 3.85iT - 97T^{2} \)
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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

−12.00142000238750676791534879592, −11.17549635048096531113886032275, −9.940963852102226416016570194842, −8.691603914089907138832683077189, −8.344497956512606405398085206830, −7.07120662505080525912573654139, −6.19043297258365006145114860564, −5.11714667388095388531819234439, −3.86401654941510180311067355455, −2.84702112699573373564978528090, 0.19801067459739994151299218809, 2.77305154333901874499403334491, 3.53282184584938337357128008789, 4.58634303283287559898194397554, 6.23081538822587567972308786966, 6.85330901235605080302873763793, 8.249389923293170996204316275084, 9.239143980036248776579389943816, 10.04210474792278453188992676444, 11.19439286132039845778863834023

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