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Properties

Label 2-370-185.82-c1-0-1
Degree $2$
Conductor $370$
Sign $-0.585 + 0.810i$
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 + (−2.35 + 0.630i)3-s + (−0.499 + 0.866i)4-s + (−0.0214 + 2.23i)5-s + (−1.72 − 1.72i)6-s + (−1.71 + 0.460i)7-s − 0.999·8-s + (2.54 − 1.46i)9-s + (−1.94 + 1.09i)10-s − 2.57i·11-s + (0.630 − 2.35i)12-s + (−0.427 + 0.739i)13-s + (−1.25 − 1.25i)14-s + (−1.35 − 5.27i)15-s + (−0.5 − 0.866i)16-s + (−0.757 + 0.437i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−1.35 + 0.364i)3-s + (−0.249 + 0.433i)4-s + (−0.00960 + 0.999i)5-s + (−0.703 − 0.703i)6-s + (−0.649 + 0.173i)7-s − 0.353·8-s + (0.847 − 0.489i)9-s + (−0.615 + 0.347i)10-s − 0.776i·11-s + (0.182 − 0.679i)12-s + (−0.118 + 0.205i)13-s + (−0.336 − 0.336i)14-s + (−0.350 − 1.36i)15-s + (−0.125 − 0.216i)16-s + (−0.183 + 0.106i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.585 + 0.810i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.585 + 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134556 - 0.263076i\)
\(L(\frac12)\) \(\approx\) \(0.134556 - 0.263076i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0214 - 2.23i)T \)
37 \( 1 + (-1.23 - 5.95i)T \)
good3 \( 1 + (2.35 - 0.630i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.71 - 0.460i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + 2.57iT - 11T^{2} \)
13 \( 1 + (0.427 - 0.739i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.757 - 0.437i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.00 + 3.76i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 3.91T + 23T^{2} \)
29 \( 1 + (1.14 + 1.14i)T + 29iT^{2} \)
31 \( 1 + (1.60 - 1.60i)T - 31iT^{2} \)
41 \( 1 + (-3.54 - 2.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (-6.26 - 6.26i)T + 47iT^{2} \)
53 \( 1 + (9.72 + 2.60i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.20 + 0.859i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.49 - 9.30i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.793 - 2.95i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.32 + 2.29i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.92 - 8.92i)T + 73iT^{2} \)
79 \( 1 + (-1.90 - 7.09i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (8.30 + 2.22i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-4.21 + 15.7i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + 2.34iT - 97T^{2} \)
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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

−11.76590937202513036981487203648, −11.20454939532783874488621730218, −10.34366193494798661394958568752, −9.409075415136095624863460668647, −8.057009318293892471375060514526, −6.74085280846900573036532582970, −6.30927600057393299498207682101, −5.44345460343513404285377379018, −4.24074329057753150154870715870, −2.95861069542295452733601320068, 0.20374894836351664066642204302, 1.79133186365126244954148827996, 3.84655908081512845237772376221, 4.93365424221084373063731911133, 5.74371754505772476076786947000, 6.65232853230979327719867232238, 7.914439227735721937593499515504, 9.300541055948564314671625351350, 10.07218927532509393239723418363, 10.97799612519703405157765482757

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