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Properties

Label 2-380-76.31-c1-0-5
Degree $2$
Conductor $380$
Sign $-0.00350 - 0.999i$
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  + (−0.840 − 1.13i)2-s + (1.52 + 2.64i)3-s + (−0.586 + 1.91i)4-s + (0.5 + 0.866i)5-s + (1.72 − 3.96i)6-s + 2.58i·7-s + (2.66 − 0.939i)8-s + (−3.16 + 5.48i)9-s + (0.564 − 1.29i)10-s + 0.0502i·11-s + (−5.95 + 1.36i)12-s + (−2.19 − 1.26i)13-s + (2.93 − 2.16i)14-s + (−1.52 + 2.64i)15-s + (−3.31 − 2.24i)16-s + (−3.71 − 6.43i)17-s + ⋯
L(s)  = 1  + (−0.594 − 0.804i)2-s + (0.882 + 1.52i)3-s + (−0.293 + 0.955i)4-s + (0.223 + 0.387i)5-s + (0.704 − 1.61i)6-s + 0.975i·7-s + (0.943 − 0.332i)8-s + (−1.05 + 1.82i)9-s + (0.178 − 0.410i)10-s + 0.0151i·11-s + (−1.71 + 0.394i)12-s + (−0.608 − 0.351i)13-s + (0.784 − 0.579i)14-s + (−0.394 + 0.683i)15-s + (−0.827 − 0.561i)16-s + (−0.901 − 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00350 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00350 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.00350 - 0.999i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.00350 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.876796 + 0.879878i\)
\(L(\frac12)\) \(\approx\) \(0.876796 + 0.879878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.840 + 1.13i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-4.32 - 0.540i)T \)
good3 \( 1 + (-1.52 - 2.64i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 2.58iT - 7T^{2} \)
11 \( 1 - 0.0502iT - 11T^{2} \)
13 \( 1 + (2.19 + 1.26i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.71 + 6.43i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.82 - 1.05i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.821 + 0.474i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.30T + 31T^{2} \)
37 \( 1 - 8.51iT - 37T^{2} \)
41 \( 1 + (-9.25 + 5.34i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.12 - 4.11i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.59 - 4.38i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.5 - 6.06i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.90 + 3.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.165 - 0.286i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.82 + 4.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.89 - 6.74i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.13 + 8.89i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.09 + 5.35i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.75iT - 83T^{2} \)
89 \( 1 + (8.19 + 4.72i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.4 - 6.05i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.33776274737053993046891203513, −10.45986390120948489783576809432, −9.539848756719756306532964832579, −9.293798175520901835008506267915, −8.349664257925384089712057241913, −7.27838445132114176001808839819, −5.33268964647522931704487262981, −4.41210083307001318128369833111, −3.00264677513363990046663176528, −2.56707689473587105201229809261, 0.958040175696273275868860932626, 2.17320836427084431615521273310, 4.12661753187123650730163854601, 5.73338305604993458459996731968, 6.84858063579455369665679447481, 7.29723367445922157984240613892, 8.245354066936078896972129163712, 8.895497081157663297702410910573, 9.883329471671839926241232198483, 11.00859804188263156871769332351

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