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Properties

Label 2-370-185.49-c1-0-2
Degree $2$
Conductor $370$
Sign $-0.833 + 0.551i$
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  + (0.642 + 0.766i)2-s + (−1.57 + 1.87i)3-s + (−0.173 + 0.984i)4-s + (−1.90 + 1.17i)5-s − 2.44·6-s + (0.511 + 1.40i)7-s + (−0.866 + 0.500i)8-s + (−0.517 − 2.93i)9-s + (−2.12 − 0.703i)10-s + (−1.13 − 1.96i)11-s + (−1.57 − 1.87i)12-s + (5.41 + 0.955i)13-s + (−0.748 + 1.29i)14-s + (0.791 − 5.41i)15-s + (−0.939 − 0.342i)16-s + (−6.62 + 1.16i)17-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (−0.907 + 1.08i)3-s + (−0.0868 + 0.492i)4-s + (−0.851 + 0.525i)5-s − 0.998·6-s + (0.193 + 0.531i)7-s + (−0.306 + 0.176i)8-s + (−0.172 − 0.978i)9-s + (−0.671 − 0.222i)10-s + (−0.341 − 0.592i)11-s + (−0.453 − 0.540i)12-s + (1.50 + 0.265i)13-s + (−0.200 + 0.346i)14-s + (0.204 − 1.39i)15-s + (−0.234 − 0.0855i)16-s + (−1.60 + 0.283i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.833 + 0.551i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.833 + 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210024 - 0.698018i\)
\(L(\frac12)\) \(\approx\) \(0.210024 - 0.698018i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 - 0.766i)T \)
5 \( 1 + (1.90 - 1.17i)T \)
37 \( 1 + (4.94 - 3.54i)T \)
good3 \( 1 + (1.57 - 1.87i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.511 - 1.40i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.13 + 1.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.41 - 0.955i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (6.62 - 1.16i)T + (15.9 - 5.81i)T^{2} \)
19 \( 1 + (6.54 + 5.49i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-4.17 - 2.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.99 - 8.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.16T + 31T^{2} \)
41 \( 1 + (1.18 - 6.70i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 5.46iT - 43T^{2} \)
47 \( 1 + (3.76 + 2.17i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.37 - 6.52i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (8.56 + 3.11i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.781 - 4.43i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-4.01 - 11.0i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-2.08 - 1.74i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 2.87iT - 73T^{2} \)
79 \( 1 + (-0.296 + 0.107i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (10.7 - 1.89i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-9.46 - 3.44i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-1.84 - 1.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.53269994867043441636557708014, −11.09181326779703819300279849530, −10.59661974889959005800468551160, −8.839305946187244888857742608393, −8.484842718195879003597309101088, −6.78539356196327854168055029079, −6.22011229263196594312204264008, −4.92891782881012027400031520686, −4.28003556131464248557224435843, −3.08085999104484047954239088272, 0.47507318214261646161731490110, 1.89754657490246129728596074484, 3.89829621442947024519418908405, 4.75398558409386302803829403676, 6.09097976511689894667578953953, 6.82207286779175064261624290787, 7.975271333394954459140243436869, 8.823385180465901501477706639035, 10.55472567769722425357072043806, 11.00568463961141980270142562256

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