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Properties

Label 2-370-37.36-c1-0-7
Degree $2$
Conductor $370$
Sign $0.525 + 0.850i$
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  + i·2-s − 1.14·3-s − 4-s − i·5-s − 1.14i·6-s − 0.342·7-s − i·8-s − 1.68·9-s + 10-s + 1.19·11-s + 1.14·12-s − 4.68i·13-s − 0.342i·14-s + 1.14i·15-s + 16-s − 4.17i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.661·3-s − 0.5·4-s − 0.447i·5-s − 0.468i·6-s − 0.129·7-s − 0.353i·8-s − 0.561·9-s + 0.316·10-s + 0.360·11-s + 0.330·12-s − 1.29i·13-s − 0.0916i·14-s + 0.295i·15-s + 0.250·16-s − 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617554 - 0.344413i\)
\(L(\frac12)\) \(\approx\) \(0.617554 - 0.344413i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + iT \)
37 \( 1 + (3.19 + 5.17i)T \)
good3 \( 1 + 1.14T + 3T^{2} \)
7 \( 1 + 0.342T + 7T^{2} \)
11 \( 1 - 1.19T + 11T^{2} \)
13 \( 1 + 4.68iT - 13T^{2} \)
17 \( 1 + 4.17iT - 17T^{2} \)
19 \( 1 + 3.14iT - 19T^{2} \)
23 \( 1 + 4.68iT - 23T^{2} \)
29 \( 1 - 0.803iT - 29T^{2} \)
31 \( 1 - 4.63iT - 31T^{2} \)
41 \( 1 - 3.78T + 41T^{2} \)
43 \( 1 + 1.19iT - 43T^{2} \)
47 \( 1 - 3.83T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 - 0.167iT - 59T^{2} \)
61 \( 1 - 2.17iT - 61T^{2} \)
67 \( 1 + 1.24T + 67T^{2} \)
71 \( 1 - 0.685T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 - 5.14iT - 79T^{2} \)
83 \( 1 - 4.22T + 83T^{2} \)
89 \( 1 + 4.58iT - 89T^{2} \)
97 \( 1 + 4.21iT - 97T^{2} \)
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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

−11.24351360592439260654377680822, −10.36339021387972400136835650220, −9.204454609720456101514015938899, −8.467945555306155161104821769906, −7.37842996367695302901964751585, −6.35135894572237812660437542987, −5.43886068796977401790629280949, −4.68472604067084461177339499677, −3.03382614885322601685958042059, −0.52537967101425068786978737934, 1.76169980122040947232616136882, 3.33995885709405898999953572755, 4.44076266777878734100764604922, 5.80038710907015613689055800045, 6.53999548683079857210553413401, 7.899910372832608648497055592126, 9.011580884845708462647618050866, 9.884372998547044949999666792093, 10.84621041198280719911883933654, 11.53950805851911405950000659779

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