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Properties

Label 2-370-185.64-c1-0-18
Degree $2$
Conductor $370$
Sign $-0.936 + 0.351i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.686 − 0.396i)3-s + (−0.499 − 0.866i)4-s + (−1.5 − 1.65i)5-s + 0.792i·6-s + (−3 + 1.73i)7-s + 0.999·8-s + (−1.18 + 2.05i)9-s + (2.18 − 0.469i)10-s + (−0.686 − 0.396i)12-s + (−2.68 − 4.65i)13-s − 3.46i·14-s + (−1.68 − 0.543i)15-s + (−0.5 + 0.866i)16-s + (−2.18 + 3.78i)17-s + (−1.18 − 2.05i)18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.396 − 0.228i)3-s + (−0.249 − 0.433i)4-s + (−0.670 − 0.741i)5-s + 0.323i·6-s + (−1.13 + 0.654i)7-s + 0.353·8-s + (−0.395 + 0.684i)9-s + (0.691 − 0.148i)10-s + (−0.198 − 0.114i)12-s + (−0.745 − 1.29i)13-s − 0.925i·14-s + (−0.435 − 0.140i)15-s + (−0.125 + 0.216i)16-s + (−0.530 + 0.918i)17-s + (−0.279 − 0.484i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.936 + 0.351i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.936 + 0.351i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 1.65i)T \)
37 \( 1 + (-0.5 + 6.06i)T \)
good3 \( 1 + (-0.686 + 0.396i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (3 - 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (2.68 + 4.65i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.18 - 3.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.74T + 23T^{2} \)
29 \( 1 + 4.40iT - 29T^{2} \)
31 \( 1 - 1.08iT - 31T^{2} \)
41 \( 1 + (2.87 + 4.97i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.11T + 43T^{2} \)
47 \( 1 - 1.58iT - 47T^{2} \)
53 \( 1 + (-3.68 - 2.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.62 + 2.67i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.55 - 3.78i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.1 + 5.84i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.37 - 7.57i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + (-10.1 + 5.84i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.255 + 0.147i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (11.1 + 6.45i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.11T + 97T^{2} \)
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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

−10.81498967256871285936363921432, −9.843890948368760744055583870273, −8.924221954480204673450960015061, −8.124948522730085121833255073320, −7.54642146519005609743315884400, −6.11740602243252145076170784437, −5.34272215297827295424750096398, −3.90912450530183781296641046838, −2.40446390556957536163841765474, 0, 2.55310379476121640911379277657, 3.55242703230157424368167477410, 4.35166462100043483663066563548, 6.43710549241103880379807921398, 7.08054401987464178557179478376, 8.201838672917920221575809326230, 9.425937517448192963360001029391, 9.782204625008507401500231957809, 10.85436413399536939595985692466

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