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Properties

Label 2-370-185.142-c1-0-6
Degree $2$
Conductor $370$
Sign $0.701 - 0.712i$
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 − i)3-s − 4-s + (2 + i)5-s + (1 − i)6-s + (1 + i)7-s − i·8-s − i·9-s + (−1 + 2i)10-s + 2i·11-s + (1 + i)12-s − 2i·13-s + (−1 + i)14-s + (−1 − 3i)15-s + 16-s + 4·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.577 − 0.577i)3-s − 0.5·4-s + (0.894 + 0.447i)5-s + (0.408 − 0.408i)6-s + (0.377 + 0.377i)7-s − 0.353i·8-s − 0.333i·9-s + (−0.316 + 0.632i)10-s + 0.603i·11-s + (0.288 + 0.288i)12-s − 0.554i·13-s + (−0.267 + 0.267i)14-s + (−0.258 − 0.774i)15-s + 0.250·16-s + 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20964 + 0.506267i\)
\(L(\frac12)\) \(\approx\) \(1.20964 + 0.506267i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - i)T \)
37 \( 1 + (6 + i)T \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (-5 - 5i)T + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 + (-7 - 7i)T + 31iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (7 + 7i)T + 47iT^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + (-1 - i)T + 59iT^{2} \)
61 \( 1 + (3 + 3i)T + 61iT^{2} \)
67 \( 1 + (3 - 3i)T - 67iT^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (9 + 9i)T + 73iT^{2} \)
79 \( 1 + (-1 - i)T + 79iT^{2} \)
83 \( 1 + (5 - 5i)T - 83iT^{2} \)
89 \( 1 + (5 - 5i)T - 89iT^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−11.85802598157211148736903372086, −10.31116028768085837893689030303, −9.819277877016164792075999364668, −8.585083425982287523215195122926, −7.51897958150786303110733817438, −6.68526356117287462089483685564, −5.77520787672965429489541320677, −5.15287970719384896425515269860, −3.28734478069087000181489345612, −1.45751200082893960204319637478, 1.22265141455069238326682312772, 2.86806587891364778727409853097, 4.47664800244323080901804971475, 5.14830422622777510526286974851, 6.13613439725025295932208442071, 7.67713971542231896259485145227, 8.823461997305896704784924605521, 9.743408199867838337470935935711, 10.34648853603495862491516881824, 11.28381770311072609603543422432

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