Properties

Label 2-370-37.34-c1-0-0
Degree $2$
Conductor $370$
Sign $-0.426 - 0.904i$
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  + (−0.766 − 0.642i)2-s + (0.204 − 0.171i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s − 0.267·6-s + (−3.03 − 1.10i)7-s + (0.500 − 0.866i)8-s + (−0.508 + 2.88i)9-s + (0.5 + 0.866i)10-s + (−0.746 + 1.29i)11-s + (0.204 + 0.171i)12-s + (0.0469 + 0.266i)13-s + (1.61 + 2.79i)14-s + (−0.251 + 0.0914i)15-s + (−0.939 + 0.342i)16-s + (−0.420 + 2.38i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.118 − 0.0992i)3-s + (0.0868 + 0.492i)4-s + (−0.420 − 0.152i)5-s − 0.109·6-s + (−1.14 − 0.417i)7-s + (0.176 − 0.306i)8-s + (−0.169 + 0.961i)9-s + (0.158 + 0.273i)10-s + (−0.225 + 0.390i)11-s + (0.0591 + 0.0496i)12-s + (0.0130 + 0.0738i)13-s + (0.431 + 0.748i)14-s + (−0.0649 + 0.0236i)15-s + (−0.234 + 0.0855i)16-s + (−0.101 + 0.578i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.426 - 0.904i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152312 + 0.240288i\)
\(L(\frac12)\) \(\approx\) \(0.152312 + 0.240288i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (5.96 - 1.21i)T \)
good3 \( 1 + (-0.204 + 0.171i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (3.03 + 1.10i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (0.746 - 1.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0469 - 0.266i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.420 - 2.38i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (5.39 - 4.52i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (1.46 + 2.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0380 - 0.0659i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.49T + 31T^{2} \)
41 \( 1 + (-0.903 - 5.12i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 1.65T + 43T^{2} \)
47 \( 1 + (-1.45 - 2.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.50 + 2.00i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-4.61 + 1.68i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.0607 - 0.344i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (6.52 + 2.37i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-3.03 + 2.54i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + 0.00494T + 73T^{2} \)
79 \( 1 + (14.6 + 5.34i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.80 + 15.9i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-3.04 + 1.11i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (5.03 + 8.72i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.61003584121297575483169777945, −10.43912642162415306967557336075, −10.19403432484772055731347513382, −8.869613478946451012293936388941, −8.103943967011308059451997200956, −7.17831830026734248946041067643, −6.10141219524105229797326624250, −4.49908468381543744642722636729, −3.41434362531056140742643895667, −2.01369836985851388069966515242, 0.21172068011346814346779111334, 2.71458353552684314818090188296, 3.89170583976855795560892704621, 5.50903321408053992361209237034, 6.50374470707694472926065055537, 7.18007006548003921708984219397, 8.567694877643010426444342305371, 9.101251853644916540930383908021, 9.988357385870321937167806632119, 10.98688641418369005389463373087

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