Properties

Label 2-370-185.68-c1-0-12
Degree $2$
Conductor $370$
Sign $0.497 + 0.867i$
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  − 2-s + (2.41 − 2.41i)3-s + 4-s + (0.667 + 2.13i)5-s + (−2.41 + 2.41i)6-s + (−0.875 + 0.875i)7-s − 8-s − 8.67i·9-s + (−0.667 − 2.13i)10-s − 1.92i·11-s + (2.41 − 2.41i)12-s + 3.05·13-s + (0.875 − 0.875i)14-s + (6.76 + 3.54i)15-s + 16-s − 3.63i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (1.39 − 1.39i)3-s + 0.5·4-s + (0.298 + 0.954i)5-s + (−0.986 + 0.986i)6-s + (−0.330 + 0.330i)7-s − 0.353·8-s − 2.89i·9-s + (−0.211 − 0.674i)10-s − 0.580i·11-s + (0.697 − 0.697i)12-s + 0.847·13-s + (0.234 − 0.234i)14-s + (1.74 + 0.914i)15-s + 0.250·16-s − 0.881i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.497 + 0.867i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.497 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36655 - 0.791423i\)
\(L(\frac12)\) \(\approx\) \(1.36655 - 0.791423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + (-0.667 - 2.13i)T \)
37 \( 1 + (0.586 - 6.05i)T \)
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 \( 1 - 3.05T + 13T^{2} \)
17 \( 1 + 3.63iT - 17T^{2} \)
19 \( 1 + (-3.22 - 3.22i)T + 19iT^{2} \)
23 \( 1 + 2.01T + 23T^{2} \)
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 \( 1 + 4.81T + 43T^{2} \)
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 \( 1 - 14.8iT - 97T^{2} \)
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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

−11.39632961886210435616673206795, −9.927624839798723273507818089646, −9.330698943183947389661457604804, −8.207838745034276224166302437561, −7.78364271742287520435035879888, −6.55295358874171615209842069297, −6.21675786827078097926275305327, −3.33037706603762223103097302954, −2.75798045642095870317445441544, −1.39077293776433931916942311112, 1.91667935302090990450380563816, 3.38776881780332949000395954211, 4.36086240291724903144583836322, 5.49091373858763078696464073964, 7.22012732960130472545939877577, 8.435964359178981519058916815179, 8.729825211211478066810805261888, 9.687651206997528340958820599110, 10.18432027993809538454165248420, 11.07488821840138530496568033577

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