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Properties

Label 2-370-5.4-c1-0-2
Degree $2$
Conductor $370$
Sign $-0.894 + 0.447i$
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 + 2.44i·3-s − 4-s + (−2 + i)5-s − 2.44·6-s + 4.44i·7-s − i·8-s − 2.99·9-s + (−1 − 2i)10-s + 4.89·11-s − 2.44i·12-s − 4i·13-s − 4.44·14-s + (−2.44 − 4.89i)15-s + 16-s − 4.89i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.41i·3-s − 0.5·4-s + (−0.894 + 0.447i)5-s − 0.999·6-s + 1.68i·7-s − 0.353i·8-s − 0.999·9-s + (−0.316 − 0.632i)10-s + 1.47·11-s − 0.707i·12-s − 1.10i·13-s − 1.18·14-s + (−0.632 − 1.26i)15-s + 0.250·16-s − 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233569 - 0.989414i\)
\(L(\frac12)\) \(\approx\) \(0.233569 - 0.989414i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 - 2.44iT - 3T^{2} \)
7 \( 1 - 4.44iT - 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 + 3.55T + 19T^{2} \)
23 \( 1 - 8.89iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 1.55T + 31T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 4.44iT - 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 - 3.55T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + 5.55iT - 67T^{2} \)
71 \( 1 - 4.89T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 6.44T + 79T^{2} \)
83 \( 1 - 9.55iT - 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−11.73315927642622350769076202571, −11.07936341415248186996918047476, −9.779503443908325277954978684317, −9.158470199486185831464631784296, −8.432623073049425086652630510554, −7.23217428890444396756726139354, −5.95292952459366795557344906314, −5.12846248971825754738029212440, −4.00228210919911626960026921576, −3.04726820625125862859212933148, 0.74264998369890049481600485601, 1.78930735016282257625297775982, 3.90734779431379043345770183706, 4.30837356157822123939895533126, 6.55394008362429058527231465249, 6.94772658743868467461326405431, 8.126851331913683768832510524418, 8.780027250126575614214292812127, 10.17220101306183106204817417116, 11.18886339174866365554162552238

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