Properties

Label 2-380-20.7-c1-0-45
Degree $2$
Conductor $380$
Sign $-0.258 + 0.965i$
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  + (0.979 − 1.01i)2-s + (−0.582 − 0.582i)3-s + (−0.0796 − 1.99i)4-s + (1.46 + 1.68i)5-s + (−1.16 + 0.0231i)6-s + (−0.972 + 0.972i)7-s + (−2.11 − 1.87i)8-s − 2.32i·9-s + (3.15 + 0.157i)10-s − 4.73i·11-s + (−1.11 + 1.21i)12-s + (4.00 − 4.00i)13-s + (0.0387 + 1.94i)14-s + (0.128 − 1.83i)15-s + (−3.98 + 0.318i)16-s + (1.22 + 1.22i)17-s + ⋯
L(s)  = 1  + (0.692 − 0.721i)2-s + (−0.336 − 0.336i)3-s + (−0.0398 − 0.999i)4-s + (0.656 + 0.754i)5-s + (−0.475 + 0.00947i)6-s + (−0.367 + 0.367i)7-s + (−0.748 − 0.663i)8-s − 0.773i·9-s + (0.998 + 0.0496i)10-s − 1.42i·11-s + (−0.322 + 0.349i)12-s + (1.11 − 1.11i)13-s + (0.0103 + 0.519i)14-s + (0.0330 − 0.474i)15-s + (−0.996 + 0.0796i)16-s + (0.297 + 0.297i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.258 + 0.965i$
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: \((2,\ 380,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10376 - 1.43816i\)
\(L(\frac12)\) \(\approx\) \(1.10376 - 1.43816i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.979 + 1.01i)T \)
5 \( 1 + (-1.46 - 1.68i)T \)
19 \( 1 + T \)
good3 \( 1 + (0.582 + 0.582i)T + 3iT^{2} \)
7 \( 1 + (0.972 - 0.972i)T - 7iT^{2} \)
11 \( 1 + 4.73iT - 11T^{2} \)
13 \( 1 + (-4.00 + 4.00i)T - 13iT^{2} \)
17 \( 1 + (-1.22 - 1.22i)T + 17iT^{2} \)
23 \( 1 + (-1.83 - 1.83i)T + 23iT^{2} \)
29 \( 1 - 6.49iT - 29T^{2} \)
31 \( 1 + 1.38iT - 31T^{2} \)
37 \( 1 + (-3.47 - 3.47i)T + 37iT^{2} \)
41 \( 1 + 6.68T + 41T^{2} \)
43 \( 1 + (-4.72 - 4.72i)T + 43iT^{2} \)
47 \( 1 + (-6.12 + 6.12i)T - 47iT^{2} \)
53 \( 1 + (-3.65 + 3.65i)T - 53iT^{2} \)
59 \( 1 - 0.289T + 59T^{2} \)
61 \( 1 + 4.63T + 61T^{2} \)
67 \( 1 + (3.20 - 3.20i)T - 67iT^{2} \)
71 \( 1 - 14.8iT - 71T^{2} \)
73 \( 1 + (-6.01 + 6.01i)T - 73iT^{2} \)
79 \( 1 + 5.24T + 79T^{2} \)
83 \( 1 + (-4.42 - 4.42i)T + 83iT^{2} \)
89 \( 1 + 8.67iT - 89T^{2} \)
97 \( 1 + (3.14 + 3.14i)T + 97iT^{2} \)
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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

−11.06026897067105116340678718907, −10.55869599100620932025989601266, −9.481312327697730054560451023163, −8.538878984333503454439902644754, −6.82681488238025445710134366086, −5.92992755004111535669958397803, −5.63056115809047436605061868616, −3.56110152591114002938407737456, −2.97489203355246319682965505161, −1.13308096658030050304865081930, 2.14866139711485868506095888771, 4.11239192123751600550281775170, 4.71863284363260243148835630229, 5.76954593683227747703041109053, 6.69725187860071518155855657122, 7.71108925403918165118701634232, 8.850776925189973048540143974953, 9.671601480958694221197772341971, 10.73300375741904768981425829913, 11.87460140330681301898198523192

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