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Properties

Label 2-370-185.104-c1-0-0
Degree $2$
Conductor $370$
Sign $-0.961 + 0.273i$
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  + (0.939 − 0.342i)2-s + (−1.06 + 2.93i)3-s + (0.766 − 0.642i)4-s + (−1.98 − 1.02i)5-s + 3.11i·6-s + (−4.22 + 0.745i)7-s + (0.500 − 0.866i)8-s + (−5.15 − 4.32i)9-s + (−2.21 − 0.288i)10-s + (1.53 − 2.66i)11-s + (1.06 + 2.93i)12-s + (−1.79 + 1.50i)13-s + (−3.71 + 2.14i)14-s + (5.13 − 4.72i)15-s + (0.173 − 0.984i)16-s + (−0.650 − 0.545i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (−0.616 + 1.69i)3-s + (0.383 − 0.321i)4-s + (−0.887 − 0.460i)5-s + 1.27i·6-s + (−1.59 + 0.281i)7-s + (0.176 − 0.306i)8-s + (−1.71 − 1.44i)9-s + (−0.701 − 0.0911i)10-s + (0.463 − 0.802i)11-s + (0.308 + 0.846i)12-s + (−0.498 + 0.418i)13-s + (−0.993 + 0.573i)14-s + (1.32 − 1.21i)15-s + (0.0434 − 0.246i)16-s + (−0.157 − 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.961 + 0.273i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.961 + 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0389789 - 0.279178i\)
\(L(\frac12)\) \(\approx\) \(0.0389789 - 0.279178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (1.98 + 1.02i)T \)
37 \( 1 + (5.56 - 2.45i)T \)
good3 \( 1 + (1.06 - 2.93i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (4.22 - 0.745i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-1.53 + 2.66i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.79 - 1.50i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.650 + 0.545i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (1.67 - 4.60i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (-2.03 - 3.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.82 + 3.36i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.08iT - 31T^{2} \)
41 \( 1 + (9.54 - 8.00i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 4.64T + 43T^{2} \)
47 \( 1 + (-0.254 + 0.146i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.92 + 0.691i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-11.5 - 2.03i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (0.829 + 0.987i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-6.59 + 1.16i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-12.1 - 4.42i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 13.9iT - 73T^{2} \)
79 \( 1 + (7.28 - 1.28i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (6.19 - 7.38i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-11.8 - 2.08i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (2.11 + 3.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.74196889268812554781434368601, −11.17061815904356123595482145882, −10.02428049696708517473741514416, −9.506788890122401485028519485595, −8.538399644491037429272566482387, −6.74433587345949250803416897644, −5.80751206103035988307575243473, −4.89007394804124128602147575934, −3.71002279377613783742477024837, −3.38972018409637676833018473189, 0.15449095819620945613979322686, 2.40639403415038290483165682342, 3.62242831560787175135006808318, 5.18838483958586578970443183099, 6.57742882818604932000985358169, 6.84839957522797503595708721052, 7.45824980014767786000761580532, 8.737654824040406142603169081140, 10.29896357138169245024356662350, 11.29740550955178706896801246683

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