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Properties

Label 2-380-95.13-c1-0-4
Degree $2$
Conductor $380$
Sign $0.837 - 0.546i$
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  + (0.959 − 0.447i)3-s + (1.57 + 1.58i)5-s + (0.204 + 0.763i)7-s + (−1.20 + 1.43i)9-s + (2.14 + 3.71i)11-s + (−1.21 + 2.61i)13-s + (2.22 + 0.811i)15-s + (0.544 − 6.22i)17-s + (0.761 − 4.29i)19-s + (0.537 + 0.641i)21-s + (2.67 − 1.87i)23-s + (−0.00791 + 4.99i)25-s + (−1.33 + 4.98i)27-s + (−6.38 − 5.35i)29-s + (4.23 + 2.44i)31-s + ⋯
L(s)  = 1  + (0.553 − 0.258i)3-s + (0.706 + 0.707i)5-s + (0.0773 + 0.288i)7-s + (−0.402 + 0.479i)9-s + (0.645 + 1.11i)11-s + (−0.337 + 0.724i)13-s + (0.574 + 0.209i)15-s + (0.132 − 1.50i)17-s + (0.174 − 0.984i)19-s + (0.117 + 0.139i)21-s + (0.558 − 0.391i)23-s + (−0.00158 + 0.999i)25-s + (−0.257 + 0.960i)27-s + (−1.18 − 0.994i)29-s + (0.760 + 0.439i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.837 - 0.546i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.837 - 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69363 + 0.503835i\)
\(L(\frac12)\) \(\approx\) \(1.69363 + 0.503835i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.57 - 1.58i)T \)
19 \( 1 + (-0.761 + 4.29i)T \)
good3 \( 1 + (-0.959 + 0.447i)T + (1.92 - 2.29i)T^{2} \)
7 \( 1 + (-0.204 - 0.763i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.14 - 3.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.21 - 2.61i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (-0.544 + 6.22i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (-2.67 + 1.87i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (6.38 + 5.35i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-4.23 - 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.75 + 1.75i)T + 37iT^{2} \)
41 \( 1 + (-0.896 + 2.46i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-5.40 + 7.71i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (-4.32 + 0.378i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (0.286 + 0.408i)T + (-18.1 + 49.8i)T^{2} \)
59 \( 1 + (9.10 - 7.64i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.00 - 5.72i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.376 + 4.30i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (15.2 - 2.68i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (5.98 + 12.8i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (-1.72 - 0.628i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-9.12 + 2.44i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (12.0 - 4.40i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (15.3 + 1.34i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.48087404552153719768071998740, −10.48628289076781510194547238836, −9.338865381821834604842637258162, −9.026764045518988489240326248547, −7.34274939458909634312955793007, −7.05544257291107081318911715451, −5.65464193000039198302527198567, −4.54215102064885074265975562458, −2.82708876601035336366364799203, −2.05932819720321792782386723387, 1.31012675662684214264334676519, 3.10253832383514062072604369632, 4.10087508418094361979217764539, 5.62665968453126791990937659934, 6.19111887414028016708598445067, 7.84251094820004742311321797171, 8.615137530431174228506240790240, 9.327570562484923315855331773917, 10.21803462819534057710904998713, 11.17885161923786864072360814761

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