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Properties

Label 2-370-37.16-c1-0-7
Degree $2$
Conductor $370$
Sign $0.988 + 0.148i$
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 + (2.77 − 1.00i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s − 2.95·6-s + (−0.530 + 3.00i)7-s + (−0.500 − 0.866i)8-s + (4.37 − 3.66i)9-s + (0.5 − 0.866i)10-s + (1.06 + 1.83i)11-s + (2.77 + 1.00i)12-s + (0.822 + 0.690i)13-s + (1.52 − 2.64i)14-s + (0.512 + 2.90i)15-s + (0.173 + 0.984i)16-s + (4.15 − 3.48i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (1.60 − 0.582i)3-s + (0.383 + 0.321i)4-s + (−0.0776 + 0.440i)5-s − 1.20·6-s + (−0.200 + 1.13i)7-s + (−0.176 − 0.306i)8-s + (1.45 − 1.22i)9-s + (0.158 − 0.273i)10-s + (0.319 + 0.553i)11-s + (0.800 + 0.291i)12-s + (0.228 + 0.191i)13-s + (0.408 − 0.707i)14-s + (0.132 + 0.750i)15-s + (0.0434 + 0.246i)16-s + (1.00 − 0.846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.988 + 0.148i$
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: \((2,\ 370,\ (\ :1/2),\ 0.988 + 0.148i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65743 - 0.123479i\)
\(L(\frac12)\) \(\approx\) \(1.65743 - 0.123479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.36 - 2.87i)T \)
good3 \( 1 + (-2.77 + 1.00i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (0.530 - 3.00i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-1.06 - 1.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.822 - 0.690i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-4.15 + 3.48i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (-1.64 + 0.600i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (0.696 - 1.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.11 + 8.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
41 \( 1 + (-0.111 - 0.0934i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 - 7.30T + 43T^{2} \)
47 \( 1 + (-3.70 + 6.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.782 - 4.43i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (1.29 + 7.34i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.41 - 1.18i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (1.62 - 9.24i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (6.84 - 2.49i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 15.7T + 73T^{2} \)
79 \( 1 + (-2.01 + 11.4i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.29 + 1.92i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-0.129 - 0.735i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (4.30 - 7.45i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.54181929178530331617151311271, −10.03957577233358143155867615145, −9.285538976697980204111292089637, −8.803891855031373393588006695045, −7.63172238831679733075050273694, −7.21126440135074485450939092713, −5.79864128465726379904429666271, −3.74315998461650312200722350954, −2.75645493917272060974379519903, −1.84565244230539308718711008582, 1.49839822815320827699035229058, 3.33106077192939322351058609714, 3.98890950041495298250886590602, 5.59161573023952369250626058978, 7.23402710502511267311417189225, 7.80952375068936195636536060122, 8.803958242521866012067639852505, 9.292602946764414476437968166059, 10.34060725658976330763333458286, 10.86202240350494838124774662709

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