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Properties

Label 2-380-380.83-c1-0-14
Degree $2$
Conductor $380$
Sign $0.329 + 0.944i$
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  + (−0.366 − 1.36i)2-s + (−2.36 − 0.633i)3-s + (−1.73 + i)4-s + (1.23 − 1.86i)5-s + 3.46i·6-s + (3.46 + 3.46i)7-s + (2 + 1.99i)8-s + (2.59 + 1.50i)9-s + (−3 − 0.999i)10-s + 1.73i·11-s + (4.73 − 1.26i)12-s + (1.36 − 0.366i)13-s + (3.46 − 5.99i)14-s + (−4.09 + 3.63i)15-s + (1.99 − 3.46i)16-s + (4.09 + 1.09i)17-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−1.36 − 0.366i)3-s + (−0.866 + 0.5i)4-s + (0.550 − 0.834i)5-s + 1.41i·6-s + (1.30 + 1.30i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.500i)9-s + (−0.948 − 0.316i)10-s + 0.522i·11-s + (1.36 − 0.366i)12-s + (0.378 − 0.101i)13-s + (0.925 − 1.60i)14-s + (−1.05 + 0.938i)15-s + (0.499 − 0.866i)16-s + (0.993 + 0.266i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.329 + 0.944i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.730540 - 0.518796i\)
\(L(\frac12)\) \(\approx\) \(0.730540 - 0.518796i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
5 \( 1 + (-1.23 + 1.86i)T \)
19 \( 1 + (2.59 + 3.5i)T \)
good3 \( 1 + (2.36 + 0.633i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-3.46 - 3.46i)T + 7iT^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (-1.36 + 0.366i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-4.09 - 1.09i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (-1.26 - 4.73i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.66iT - 31T^{2} \)
37 \( 1 + (-2 + 2i)T - 37iT^{2} \)
41 \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.73 + 1.26i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-11.8 + 3.16i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.09 + 1.09i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.36 - 0.633i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.5 + 0.866i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.92 - 10.9i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.33 + 7.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.92 + 6.92i)T - 83iT^{2} \)
89 \( 1 + (0.866 + 0.5i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.46 + 1.46i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.52924200937578816043790958008, −10.51319336011330038030000876855, −9.420983551325711214031735146448, −8.619112123247560389462852090012, −7.66369343652411687010580786648, −5.87435211155103522092203689032, −5.32029816331612412471456409615, −4.46930757873736225598182002418, −2.23564413369130839083879868085, −1.15353064925330880371851115560, 1.09468076754599937284568282782, 3.95743138411987513109603953640, 4.99082182462089597280116828847, 5.82358760543928248024885164765, 6.69233895726361220595455291453, 7.54620206228658867090595195722, 8.582927567735539761531789596122, 10.11619065837160719269208925697, 10.58160449239254256355711432968, 11.05826513834453006747734795061

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