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Properties

Label 2-370-37.26-c1-0-5
Degree $2$
Conductor $370$
Sign $0.805 + 0.592i$
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  + (−0.5 − 0.866i)2-s + (0.144 − 0.250i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s − 0.289·6-s + (−1.26 + 2.18i)7-s + 0.999·8-s + (1.45 + 2.52i)9-s − 0.999·10-s + 3.81·11-s + (0.144 + 0.250i)12-s + (2.86 − 4.96i)13-s + 2.52·14-s + (−0.144 − 0.250i)15-s + (−0.5 − 0.866i)16-s + (1.84 + 3.18i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.0834 − 0.144i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s − 0.118·6-s + (−0.477 + 0.826i)7-s + 0.353·8-s + (0.486 + 0.841i)9-s − 0.316·10-s + 1.14·11-s + (0.0417 + 0.0722i)12-s + (0.794 − 1.37i)13-s + 0.674·14-s + (−0.0373 − 0.0646i)15-s + (−0.125 − 0.216i)16-s + (0.446 + 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19455 - 0.392191i\)
\(L(\frac12)\) \(\approx\) \(1.19455 - 0.392191i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-5.15 + 3.23i)T \)
good3 \( 1 + (-0.144 + 0.250i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.26 - 2.18i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 3.81T + 11T^{2} \)
13 \( 1 + (-2.86 + 4.96i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.84 - 3.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.64 + 2.84i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.81T + 23T^{2} \)
29 \( 1 - 4.20T + 29T^{2} \)
31 \( 1 - 2.44T + 31T^{2} \)
41 \( 1 + (2.14 - 3.71i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 1.33T + 43T^{2} \)
47 \( 1 + 5.64T + 47T^{2} \)
53 \( 1 + (4.50 + 7.81i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.38 - 2.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.89 + 3.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.33 - 9.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.23 - 5.60i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.68T + 73T^{2} \)
79 \( 1 + (4.81 - 8.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.91 + 5.05i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.06 - 7.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−11.31441376524831375384927710456, −10.26554674281918097371873235463, −9.599923605101825963861338388921, −8.535218001079716934709381165942, −7.949166636999463076145621305883, −6.45511118436561752728484633454, −5.46510240700244332980823877952, −4.07845937619389538444402386742, −2.78284069735148578753788491084, −1.35353940923775450100826079347, 1.30779729684796337297059234854, 3.57125942676901183339414649271, 4.37583017006602314302165414779, 6.23509721058419002339828757510, 6.58851666713846557459954857287, 7.56691398108478970641444149011, 8.879932980236489986910520564877, 9.643736220553858693288606754795, 10.19304443816871801402793756513, 11.49460136119246568161317387677

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