Properties

Label 2-370-185.142-c1-0-18
Degree $2$
Conductor $370$
Sign $0.521 - 0.853i$
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 + (−2.41 − 2.41i)3-s − 4-s + (−2.13 − 0.667i)5-s + (−2.41 + 2.41i)6-s + (−0.875 − 0.875i)7-s + i·8-s + 8.67i·9-s + (−0.667 + 2.13i)10-s − 1.92i·11-s + (2.41 + 2.41i)12-s − 3.05i·13-s + (−0.875 + 0.875i)14-s + (3.54 + 6.76i)15-s + 16-s + 3.63·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−1.39 − 1.39i)3-s − 0.5·4-s + (−0.954 − 0.298i)5-s + (−0.986 + 0.986i)6-s + (−0.330 − 0.330i)7-s + 0.353i·8-s + 2.89i·9-s + (−0.211 + 0.674i)10-s − 0.580i·11-s + (0.697 + 0.697i)12-s − 0.847i·13-s + (−0.234 + 0.234i)14-s + (0.914 + 1.74i)15-s + 0.250·16-s + 0.881·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.521 - 0.853i$
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: \((2,\ 370,\ (\ :1/2),\ 0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0967265 + 0.0542154i\)
\(L(\frac12)\) \(\approx\) \(0.0967265 + 0.0542154i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 + 0.667i)T \)
37 \( 1 + (6.05 + 0.586i)T \)
good3 \( 1 + (2.41 + 2.41i)T + 3iT^{2} \)
7 \( 1 + (0.875 + 0.875i)T + 7iT^{2} \)
11 \( 1 + 1.92iT - 11T^{2} \)
13 \( 1 + 3.05iT - 13T^{2} \)
17 \( 1 - 3.63T + 17T^{2} \)
19 \( 1 + (3.22 + 3.22i)T + 19iT^{2} \)
23 \( 1 - 2.01iT - 23T^{2} \)
29 \( 1 + (4.81 - 4.81i)T - 29iT^{2} \)
31 \( 1 + (-0.936 - 0.936i)T + 31iT^{2} \)
41 \( 1 - 12.7iT - 41T^{2} \)
43 \( 1 - 4.81iT - 43T^{2} \)
47 \( 1 + (5.85 + 5.85i)T + 47iT^{2} \)
53 \( 1 + (3.89 - 3.89i)T - 53iT^{2} \)
59 \( 1 + (-3.25 - 3.25i)T + 59iT^{2} \)
61 \( 1 + (-3.16 - 3.16i)T + 61iT^{2} \)
67 \( 1 + (-3.71 + 3.71i)T - 67iT^{2} \)
71 \( 1 + 3.90T + 71T^{2} \)
73 \( 1 + (8.57 + 8.57i)T + 73iT^{2} \)
79 \( 1 + (4.19 + 4.19i)T + 79iT^{2} \)
83 \( 1 + (7.79 - 7.79i)T - 83iT^{2} \)
89 \( 1 + (-7.79 + 7.79i)T - 89iT^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−11.04601895814673865360507600119, −10.18762416241780650984029280722, −8.506220546972071642508319987155, −7.72020826932924738952921987279, −6.84802092522255518180187141007, −5.68735380274846226631989566046, −4.78934500011627754000403295570, −3.18342072289039293976221823455, −1.29340164760864806089322263104, −0.099964061267268606280460614837, 3.68718002703902003142302175950, 4.32911772025246305387045919305, 5.39452908982263311985704481102, 6.30422722154064298695492270836, 7.17352653075638130586523247309, 8.583933543682219284610753455959, 9.589441152419568503584992915549, 10.29431880367502759113590734797, 11.20225807315990791349380899646, 12.06762619025646684483073582668

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