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Properties

Label 2-380-20.7-c1-0-52
Degree $2$
Conductor $380$
Sign $-0.866 - 0.498i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
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  + (0.433 − 1.34i)2-s + (0.497 + 0.497i)3-s + (−1.62 − 1.16i)4-s + (−2.01 − 0.978i)5-s + (0.885 − 0.454i)6-s + (−2.91 + 2.91i)7-s + (−2.27 + 1.68i)8-s − 2.50i·9-s + (−2.18 + 2.28i)10-s − 3.58i·11-s + (−0.227 − 1.38i)12-s + (−3.29 + 3.29i)13-s + (2.66 + 5.19i)14-s + (−0.514 − 1.48i)15-s + (1.27 + 3.79i)16-s + (−4.60 − 4.60i)17-s + ⋯
L(s)  = 1  + (0.306 − 0.951i)2-s + (0.287 + 0.287i)3-s + (−0.812 − 0.583i)4-s + (−0.899 − 0.437i)5-s + (0.361 − 0.185i)6-s + (−1.10 + 1.10i)7-s + (−0.804 + 0.594i)8-s − 0.834i·9-s + (−0.691 + 0.721i)10-s − 1.08i·11-s + (−0.0658 − 0.401i)12-s + (−0.914 + 0.914i)13-s + (0.712 + 1.38i)14-s + (−0.132 − 0.384i)15-s + (0.319 + 0.947i)16-s + (−1.11 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.866 - 0.498i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.866 - 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0982257 + 0.367782i\)
\(L(\frac12)\) \(\approx\) \(0.0982257 + 0.367782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.433 + 1.34i)T \)
5 \( 1 + (2.01 + 0.978i)T \)
19 \( 1 + T \)
good3 \( 1 + (-0.497 - 0.497i)T + 3iT^{2} \)
7 \( 1 + (2.91 - 2.91i)T - 7iT^{2} \)
11 \( 1 + 3.58iT - 11T^{2} \)
13 \( 1 + (3.29 - 3.29i)T - 13iT^{2} \)
17 \( 1 + (4.60 + 4.60i)T + 17iT^{2} \)
23 \( 1 + (-3.51 - 3.51i)T + 23iT^{2} \)
29 \( 1 + 4.88iT - 29T^{2} \)
31 \( 1 + 1.57iT - 31T^{2} \)
37 \( 1 + (5.48 + 5.48i)T + 37iT^{2} \)
41 \( 1 + 3.69T + 41T^{2} \)
43 \( 1 + (-4.56 - 4.56i)T + 43iT^{2} \)
47 \( 1 + (-1.60 + 1.60i)T - 47iT^{2} \)
53 \( 1 + (-2.12 + 2.12i)T - 53iT^{2} \)
59 \( 1 + 5.91T + 59T^{2} \)
61 \( 1 - 0.536T + 61T^{2} \)
67 \( 1 + (-1.64 + 1.64i)T - 67iT^{2} \)
71 \( 1 - 4.08iT - 71T^{2} \)
73 \( 1 + (0.480 - 0.480i)T - 73iT^{2} \)
79 \( 1 - 3.35T + 79T^{2} \)
83 \( 1 + (12.4 + 12.4i)T + 83iT^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 + (6.21 + 6.21i)T + 97iT^{2} \)
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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

−11.17017080806762148616092574431, −9.638612703452057604104760311166, −9.172060303570476698673659286026, −8.612015523805383508557256222433, −6.89945833763180300365639088155, −5.74074711060240252081520162035, −4.54391820042659270778134647696, −3.49759308970697710549361512331, −2.61329434974595857038640734447, −0.21250072696902941684987762596, 2.89047460264024855989615345418, 4.05682173374237597568648836963, 4.94318815832860417289725973586, 6.67613263514807765907788699331, 7.08717285676843275189770675124, 7.84616585021370637850952364010, 8.798551984521010226568325528331, 10.18876341614787356509278844434, 10.68350111010720150812562794598, 12.43115890292766733646747476628

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