Properties

Label 2-380-95.18-c1-0-0
Degree $2$
Conductor $380$
Sign $-0.658 - 0.752i$
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  + (−2.13 − 0.656i)5-s + (−1.25 + 1.25i)7-s + 3i·9-s − 2.15·11-s + (−4.25 + 4.25i)17-s + 4.35i·19-s + (−2.35 − 2.35i)23-s + (4.13 + 2.80i)25-s + (3.49 − 1.85i)35-s + (−9.11 − 9.11i)43-s + (1.97 − 6.41i)45-s + (−0.598 + 0.598i)47-s + 3.86i·49-s + (4.59 + 1.41i)55-s + 15.1·61-s + ⋯
L(s)  = 1  + (−0.955 − 0.293i)5-s + (−0.472 + 0.472i)7-s + i·9-s − 0.648·11-s + (−1.03 + 1.03i)17-s + 0.999i·19-s + (−0.491 − 0.491i)23-s + (0.827 + 0.561i)25-s + (0.590 − 0.313i)35-s + (−1.38 − 1.38i)43-s + (0.293 − 0.955i)45-s + (−0.0872 + 0.0872i)47-s + 0.552i·49-s + (0.619 + 0.190i)55-s + 1.94·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.658 - 0.752i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.218742 + 0.482254i\)
\(L(\frac12)\) \(\approx\) \(0.218742 + 0.482254i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 + 0.656i)T \)
19 \( 1 - 4.35iT \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 + (1.25 - 1.25i)T - 7iT^{2} \)
11 \( 1 + 2.15T + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (4.25 - 4.25i)T - 17iT^{2} \)
23 \( 1 + (2.35 + 2.35i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (9.11 + 9.11i)T + 43iT^{2} \)
47 \( 1 + (0.598 - 0.598i)T - 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-9.90 - 9.90i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (12.3 + 12.3i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−11.68678235770602952466475660534, −10.77667576657823109233472268668, −10.02778714541022916781271437889, −8.587359192338529654502368934088, −8.189666023945062182153096949408, −7.09107527468521810973679540380, −5.86102212031732565376695014509, −4.76697768233830865133282292117, −3.67132112516739435219401657358, −2.18421679116538324788114170284, 0.33759196080359066499717720699, 2.83111295300408771309571828225, 3.85333962976209164995910296037, 4.96089297482312004994197015263, 6.55280774516504492278983126131, 7.10404338114066028292793574502, 8.189071827400135772121139953979, 9.225846176886124372484214789410, 10.09800800895611092943880634344, 11.23394818410256994089280164797

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