Properties

Label 2-380-20.7-c1-0-8
Degree $2$
Conductor $380$
Sign $0.659 - 0.751i$
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  + (−1.34 + 0.435i)2-s + (−1.82 − 1.82i)3-s + (1.61 − 1.17i)4-s + (2.11 + 0.724i)5-s + (3.24 + 1.65i)6-s + (−1.36 + 1.36i)7-s + (−1.66 + 2.28i)8-s + 3.63i·9-s + (−3.16 − 0.0526i)10-s + 1.55i·11-s + (−5.08 − 0.814i)12-s + (−2.11 + 2.11i)13-s + (1.23 − 2.42i)14-s + (−2.53 − 5.17i)15-s + (1.24 − 3.80i)16-s + (0.503 + 0.503i)17-s + ⋯
L(s)  = 1  + (−0.951 + 0.308i)2-s + (−1.05 − 1.05i)3-s + (0.809 − 0.586i)4-s + (0.946 + 0.324i)5-s + (1.32 + 0.676i)6-s + (−0.514 + 0.514i)7-s + (−0.589 + 0.807i)8-s + 1.21i·9-s + (−0.999 − 0.0166i)10-s + 0.468i·11-s + (−1.46 − 0.235i)12-s + (−0.585 + 0.585i)13-s + (0.330 − 0.647i)14-s + (−0.654 − 1.33i)15-s + (0.312 − 0.950i)16-s + (0.122 + 0.122i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.659 - 0.751i$
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: \((2,\ 380,\ (\ :1/2),\ 0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540544 + 0.244720i\)
\(L(\frac12)\) \(\approx\) \(0.540544 + 0.244720i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.435i)T \)
5 \( 1 + (-2.11 - 0.724i)T \)
19 \( 1 - T \)
good3 \( 1 + (1.82 + 1.82i)T + 3iT^{2} \)
7 \( 1 + (1.36 - 1.36i)T - 7iT^{2} \)
11 \( 1 - 1.55iT - 11T^{2} \)
13 \( 1 + (2.11 - 2.11i)T - 13iT^{2} \)
17 \( 1 + (-0.503 - 0.503i)T + 17iT^{2} \)
23 \( 1 + (-3.79 - 3.79i)T + 23iT^{2} \)
29 \( 1 - 2.83iT - 29T^{2} \)
31 \( 1 - 4.40iT - 31T^{2} \)
37 \( 1 + (2.89 + 2.89i)T + 37iT^{2} \)
41 \( 1 - 5.57T + 41T^{2} \)
43 \( 1 + (-5.10 - 5.10i)T + 43iT^{2} \)
47 \( 1 + (-9.15 + 9.15i)T - 47iT^{2} \)
53 \( 1 + (-5.52 + 5.52i)T - 53iT^{2} \)
59 \( 1 - 9.43T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + (0.237 - 0.237i)T - 67iT^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (7.73 - 7.73i)T - 73iT^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + (0.972 + 0.972i)T + 83iT^{2} \)
89 \( 1 + 4.55iT - 89T^{2} \)
97 \( 1 + (-13.9 - 13.9i)T + 97iT^{2} \)
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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

−11.41484226091619580896091336510, −10.52563777027377352887530324583, −9.616999221087591951140997745116, −8.862708765061643082141309583753, −7.23825880335371362995012522831, −6.97316945925658732461024320505, −5.91976188499105554127201713532, −5.33094620717414593652187847195, −2.56724174611188217852254990786, −1.39483051276620983166055076647, 0.65350992862541955500875516879, 2.77248607563384023469341542685, 4.26264231437868872366386959858, 5.54441053202502394487149512830, 6.31509259975305246378494101152, 7.51881583218706259837608214249, 8.938591317257261999329391013893, 9.606058368020511490606240021531, 10.41880530297841172551829334069, 10.73165617971623919509051688521

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