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Properties

Label 2-370-37.16-c1-0-12
Degree $2$
Conductor $370$
Sign $0.225 + 0.974i$
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 + (2.25 − 0.821i)3-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)5-s − 2.40·6-s + (0.201 − 1.14i)7-s + (−0.500 − 0.866i)8-s + (2.11 − 1.77i)9-s + (−0.5 + 0.866i)10-s + (−2.27 − 3.94i)11-s + (2.25 + 0.821i)12-s + (0.690 + 0.579i)13-s + (−0.579 + 1.00i)14-s + (−0.416 − 2.36i)15-s + (0.173 + 0.984i)16-s + (−0.622 + 0.521i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (1.30 − 0.474i)3-s + (0.383 + 0.321i)4-s + (0.0776 − 0.440i)5-s − 0.980·6-s + (0.0760 − 0.431i)7-s + (−0.176 − 0.306i)8-s + (0.705 − 0.592i)9-s + (−0.158 + 0.273i)10-s + (−0.686 − 1.18i)11-s + (0.651 + 0.237i)12-s + (0.191 + 0.160i)13-s + (−0.154 + 0.268i)14-s + (−0.107 − 0.610i)15-s + (0.0434 + 0.246i)16-s + (−0.150 + 0.126i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.225 + 0.974i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.225 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16227 - 0.923884i\)
\(L(\frac12)\) \(\approx\) \(1.16227 - 0.923884i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + (-4.64 - 3.93i)T \)
good3 \( 1 + (-2.25 + 0.821i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.201 + 1.14i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (2.27 + 3.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.690 - 0.579i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.622 - 0.521i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (-5.58 + 2.03i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (1.45 - 2.52i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.06 - 5.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.34T + 31T^{2} \)
41 \( 1 + (3.23 + 2.71i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + 6.60T + 43T^{2} \)
47 \( 1 + (-2.37 + 4.11i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.718 + 4.07i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (0.0686 + 0.389i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.73 - 1.45i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (1.34 - 7.62i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-7.78 + 2.83i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 8.59T + 73T^{2} \)
79 \( 1 + (0.991 - 5.62i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (5.50 - 4.62i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-1.19 - 6.77i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (5.13 - 8.89i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.11835108687619691627305885300, −10.09218611482727920815175161646, −9.130375271692484019989082408027, −8.428430074125292697965857096697, −7.81701324385192942084767849590, −6.84812739733139811936890282638, −5.34565755503349626895016840140, −3.62299369458163381352703579744, −2.69719722064179375024510589183, −1.20107262393097438248240051846, 2.14634990313551078252900100615, 3.07905515091158518983084017272, 4.54580281549661494271976761769, 5.92865586238267517682824324013, 7.29112807203432544169433414501, 7.970264712798485086430205147819, 8.815432627551330388803268152100, 9.860118548264835565371551583413, 10.07141082758342190872017744986, 11.41858400359120096773844622906

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