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Properties

Label 2-370-37.16-c1-0-3
Degree $2$
Conductor $370$
Sign $-0.209 - 0.977i$
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 + (−1.33 + 0.484i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s − 1.41·6-s + (0.162 − 0.919i)7-s + (0.500 + 0.866i)8-s + (−0.762 + 0.640i)9-s + (−0.5 + 0.866i)10-s + (2.17 + 3.76i)11-s + (−1.33 − 0.484i)12-s + (2.10 + 1.76i)13-s + (0.467 − 0.808i)14-s + (−0.245 − 1.39i)15-s + (0.173 + 0.984i)16-s + (−5.05 + 4.24i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (−0.768 + 0.279i)3-s + (0.383 + 0.321i)4-s + (−0.0776 + 0.440i)5-s − 0.577·6-s + (0.0613 − 0.347i)7-s + (0.176 + 0.306i)8-s + (−0.254 + 0.213i)9-s + (−0.158 + 0.273i)10-s + (0.654 + 1.13i)11-s + (−0.384 − 0.139i)12-s + (0.583 + 0.489i)13-s + (0.124 − 0.216i)14-s + (−0.0634 − 0.359i)15-s + (0.0434 + 0.246i)16-s + (−1.22 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.209 - 0.977i$
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: \((2,\ 370,\ (\ :1/2),\ -0.209 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.884005 + 1.09319i\)
\(L(\frac12)\) \(\approx\) \(0.884005 + 1.09319i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.57 + 4.00i)T \)
good3 \( 1 + (1.33 - 0.484i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.162 + 0.919i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-2.17 - 3.76i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.10 - 1.76i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (5.05 - 4.24i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (1.18 - 0.430i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (0.408 - 0.708i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.40 + 4.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.21T + 31T^{2} \)
41 \( 1 + (-7.00 - 5.88i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 - 8.15T + 43T^{2} \)
47 \( 1 + (-0.927 + 1.60i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.04 + 11.6i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-1.25 - 7.14i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.93 - 1.62i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.53 + 14.4i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-6.86 + 2.49i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 16.7T + 73T^{2} \)
79 \( 1 + (-1.42 + 8.06i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.77 + 1.49i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (0.313 + 1.77i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-0.270 + 0.468i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.57749725824914813853870035649, −10.96039980775103019355526120010, −10.14125761460424680497224156017, −8.854377253386188393131243928004, −7.65815231835983106466000943844, −6.57441389076627017365853556408, −6.00890788812834805148912489521, −4.59149815186682989114547097424, −3.99121734092660535446796263468, −2.16072255547373914878084212878, 0.875986684734501093061373476334, 2.82627836280249252330298795443, 4.17007455326288454999894832754, 5.39456205003054783000657040906, 6.07740010785372391642814461181, 6.97012564596911990295299670663, 8.532216639300141361842024853363, 9.164738790327178140735255233859, 10.72328430992009187380571714540, 11.31693036568733192232318180236

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