Properties

Label 2-380-20.7-c1-0-7
Degree $2$
Conductor $380$
Sign $-0.998 - 0.0628i$
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  + (0.655 + 1.25i)2-s + (−0.124 − 0.124i)3-s + (−1.14 + 1.64i)4-s + (−0.0211 + 2.23i)5-s + (0.0741 − 0.236i)6-s + (−2.60 + 2.60i)7-s + (−2.80 − 0.352i)8-s − 2.96i·9-s + (−2.81 + 1.43i)10-s − 1.59i·11-s + (0.345 − 0.0622i)12-s + (−1.95 + 1.95i)13-s + (−4.97 − 1.55i)14-s + (0.280 − 0.274i)15-s + (−1.39 − 3.74i)16-s + (1.63 + 1.63i)17-s + ⋯
L(s)  = 1  + (0.463 + 0.886i)2-s + (−0.0716 − 0.0716i)3-s + (−0.570 + 0.821i)4-s + (−0.00945 + 0.999i)5-s + (0.0302 − 0.0966i)6-s + (−0.985 + 0.985i)7-s + (−0.992 − 0.124i)8-s − 0.989i·9-s + (−0.890 + 0.455i)10-s − 0.479i·11-s + (0.0997 − 0.0179i)12-s + (−0.541 + 0.541i)13-s + (−1.33 − 0.416i)14-s + (0.0723 − 0.0709i)15-s + (−0.349 − 0.937i)16-s + (0.397 + 0.397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.998 - 0.0628i$
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: \((2,\ 380,\ (\ :1/2),\ -0.998 - 0.0628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0346252 + 1.10078i\)
\(L(\frac12)\) \(\approx\) \(0.0346252 + 1.10078i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.655 - 1.25i)T \)
5 \( 1 + (0.0211 - 2.23i)T \)
19 \( 1 - T \)
good3 \( 1 + (0.124 + 0.124i)T + 3iT^{2} \)
7 \( 1 + (2.60 - 2.60i)T - 7iT^{2} \)
11 \( 1 + 1.59iT - 11T^{2} \)
13 \( 1 + (1.95 - 1.95i)T - 13iT^{2} \)
17 \( 1 + (-1.63 - 1.63i)T + 17iT^{2} \)
23 \( 1 + (-3.32 - 3.32i)T + 23iT^{2} \)
29 \( 1 - 2.83iT - 29T^{2} \)
31 \( 1 - 7.48iT - 31T^{2} \)
37 \( 1 + (0.620 + 0.620i)T + 37iT^{2} \)
41 \( 1 - 4.49T + 41T^{2} \)
43 \( 1 + (-4.87 - 4.87i)T + 43iT^{2} \)
47 \( 1 + (9.05 - 9.05i)T - 47iT^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 - 0.166T + 59T^{2} \)
61 \( 1 - 0.0704T + 61T^{2} \)
67 \( 1 + (-6.16 + 6.16i)T - 67iT^{2} \)
71 \( 1 + 8.99iT - 71T^{2} \)
73 \( 1 + (-9.79 + 9.79i)T - 73iT^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + (-4.60 - 4.60i)T + 83iT^{2} \)
89 \( 1 - 10.8iT - 89T^{2} \)
97 \( 1 + (-3.24 - 3.24i)T + 97iT^{2} \)
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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

−12.13811530947907023839254299979, −11.04870648562068751750490733989, −9.559252051728116954033562177371, −9.169364988292137618602757972073, −7.85329284767464713738186827361, −6.71058588539329066902012065347, −6.32355142780928185742414325227, −5.29447742426248756469096967016, −3.60043409417932222945394018705, −2.92938949996759086831385299701, 0.63545699423906737182974817248, 2.42371958797641180037640589843, 3.87541508089107361659299172798, 4.78846709159650848769568976085, 5.66033387378790937612842284989, 7.12006764901243987282188429738, 8.234823520274645786236375309690, 9.548949727563372933697404527168, 9.985325454087283846694033254376, 10.87928503185248609732037276584

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