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Properties

Label 2-370-37.36-c1-0-4
Degree $2$
Conductor $370$
Sign $-0.479 + 0.877i$
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 − 3.10·3-s − 4-s + i·5-s + 3.10i·6-s + 3.81·7-s + i·8-s + 6.62·9-s + 10-s − 4.91·11-s + 3.10·12-s − 3.62i·13-s − 3.81i·14-s − 3.10i·15-s + 16-s − 6.33i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.79·3-s − 0.5·4-s + 0.447i·5-s + 1.26i·6-s + 1.44·7-s + 0.353i·8-s + 2.20·9-s + 0.316·10-s − 1.48·11-s + 0.895·12-s − 1.00i·13-s − 1.01i·14-s − 0.801i·15-s + 0.250·16-s − 1.53i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.479 + 0.877i$
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: \((2,\ 370,\ (\ :1/2),\ -0.479 + 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.316475 - 0.533530i\)
\(L(\frac12)\) \(\approx\) \(0.316475 - 0.533530i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.91 + 5.33i)T \)
good3 \( 1 + 3.10T + 3T^{2} \)
7 \( 1 - 3.81T + 7T^{2} \)
11 \( 1 + 4.91T + 11T^{2} \)
13 \( 1 + 3.62iT - 13T^{2} \)
17 \( 1 + 6.33iT - 17T^{2} \)
19 \( 1 - 5.10iT - 19T^{2} \)
23 \( 1 + 3.62iT - 23T^{2} \)
29 \( 1 + 6.91iT - 29T^{2} \)
31 \( 1 + 4.39iT - 31T^{2} \)
41 \( 1 - 5.49T + 41T^{2} \)
43 \( 1 + 4.91iT - 43T^{2} \)
47 \( 1 + 2.52T + 47T^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
59 \( 1 + 6.52iT - 59T^{2} \)
61 \( 1 - 8.33iT - 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + 7.62T + 71T^{2} \)
73 \( 1 + 3.15T + 73T^{2} \)
79 \( 1 + 7.10iT - 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 - 12.4iT - 89T^{2} \)
97 \( 1 - 2.50iT - 97T^{2} \)
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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

−11.14461613054728385379678841371, −10.51013538419584568348684618825, −9.885020748919131335432174170338, −8.058767809257064262936362077618, −7.41880249886794775960413903521, −5.80410064147084335077337122192, −5.25489651537270955427242124003, −4.34378041327204859113369277379, −2.38149182456606463498627883928, −0.57253874518889279046480561353, 1.44721676661157891359357735599, 4.44689520430763720914960289870, 5.00624569012031963966616369741, 5.72554692809842502880262909603, 6.83604696966976962959528422791, 7.77088713300587005319312080459, 8.730665851077557588114303916424, 10.11952955718796191752854520179, 10.99334284549834502875723172831, 11.48131579491141587861620989376

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