Properties

Label 2-380-20.7-c1-0-38
Degree $2$
Conductor $380$
Sign $0.847 + 0.530i$
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  + (0.0413 + 1.41i)2-s + (0.135 + 0.135i)3-s + (−1.99 + 0.116i)4-s + (−0.869 − 2.06i)5-s + (−0.185 + 0.196i)6-s + (−0.485 + 0.485i)7-s + (−0.247 − 2.81i)8-s − 2.96i·9-s + (2.87 − 1.31i)10-s − 2.63i·11-s + (−0.285 − 0.253i)12-s + (1.08 − 1.08i)13-s + (−0.706 − 0.666i)14-s + (0.160 − 0.395i)15-s + (3.97 − 0.466i)16-s + (0.183 + 0.183i)17-s + ⋯
L(s)  = 1  + (0.0292 + 0.999i)2-s + (0.0779 + 0.0779i)3-s + (−0.998 + 0.0584i)4-s + (−0.388 − 0.921i)5-s + (−0.0756 + 0.0802i)6-s + (−0.183 + 0.183i)7-s + (−0.0875 − 0.996i)8-s − 0.987i·9-s + (0.909 − 0.415i)10-s − 0.795i·11-s + (−0.0824 − 0.0733i)12-s + (0.302 − 0.302i)13-s + (−0.188 − 0.178i)14-s + (0.0415 − 0.102i)15-s + (0.993 − 0.116i)16-s + (0.0444 + 0.0444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.847 + 0.530i$
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: \((2,\ 380,\ (\ :1/2),\ 0.847 + 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.930806 - 0.267459i\)
\(L(\frac12)\) \(\approx\) \(0.930806 - 0.267459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0413 - 1.41i)T \)
5 \( 1 + (0.869 + 2.06i)T \)
19 \( 1 - T \)
good3 \( 1 + (-0.135 - 0.135i)T + 3iT^{2} \)
7 \( 1 + (0.485 - 0.485i)T - 7iT^{2} \)
11 \( 1 + 2.63iT - 11T^{2} \)
13 \( 1 + (-1.08 + 1.08i)T - 13iT^{2} \)
17 \( 1 + (-0.183 - 0.183i)T + 17iT^{2} \)
23 \( 1 + (1.67 + 1.67i)T + 23iT^{2} \)
29 \( 1 + 2.28iT - 29T^{2} \)
31 \( 1 + 8.70iT - 31T^{2} \)
37 \( 1 + (-2.41 - 2.41i)T + 37iT^{2} \)
41 \( 1 + 6.45T + 41T^{2} \)
43 \( 1 + (-1.57 - 1.57i)T + 43iT^{2} \)
47 \( 1 + (-1.50 + 1.50i)T - 47iT^{2} \)
53 \( 1 + (-2.01 + 2.01i)T - 53iT^{2} \)
59 \( 1 - 7.70T + 59T^{2} \)
61 \( 1 + 5.31T + 61T^{2} \)
67 \( 1 + (2.32 - 2.32i)T - 67iT^{2} \)
71 \( 1 + 8.57iT - 71T^{2} \)
73 \( 1 + (-3.37 + 3.37i)T - 73iT^{2} \)
79 \( 1 + 8.20T + 79T^{2} \)
83 \( 1 + (-7.64 - 7.64i)T + 83iT^{2} \)
89 \( 1 + 7.85iT - 89T^{2} \)
97 \( 1 + (-10.0 - 10.0i)T + 97iT^{2} \)
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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

−11.47882445557108014301256521739, −9.974044120193574384882802581477, −9.159813188701924448173877422288, −8.459568486345246300103567715454, −7.66623668633879164346313665621, −6.33520971026731295205605840771, −5.65274015991554793274101737111, −4.39944924258423352876486617368, −3.45132953635735091104409514080, −0.66477265040757345616254822438, 1.89983190081227494261128824451, 3.10345221525375226799305656709, 4.20451573309485812742726452478, 5.34183494941211974246288230738, 6.86720050325159399346407267700, 7.76897077544739109460266165756, 8.798992567985429300497485407092, 10.01390751999929479702500745696, 10.52183325304400549249853949436, 11.37390731204878425115765162040

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