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Properties

Label 2-380-380.7-c1-0-51
Degree $2$
Conductor $380$
Sign $-0.996 - 0.0862i$
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  + (−1.33 + 0.470i)2-s + (0.879 − 3.28i)3-s + (1.55 − 1.25i)4-s + (−1.53 − 1.62i)5-s + (0.372 + 4.79i)6-s + (−1.21 + 1.21i)7-s + (−1.48 + 2.40i)8-s + (−7.40 − 4.27i)9-s + (2.81 + 1.44i)10-s − 2.37i·11-s + (−2.75 − 6.21i)12-s + (0.816 + 3.04i)13-s + (1.05 − 2.19i)14-s + (−6.68 + 3.61i)15-s + (0.847 − 3.90i)16-s + (0.0115 − 0.0430i)17-s + ⋯
L(s)  = 1  + (−0.942 + 0.332i)2-s + (0.507 − 1.89i)3-s + (0.778 − 0.627i)4-s + (−0.686 − 0.726i)5-s + (0.151 + 1.95i)6-s + (−0.460 + 0.460i)7-s + (−0.525 + 0.851i)8-s + (−2.46 − 1.42i)9-s + (0.889 + 0.456i)10-s − 0.716i·11-s + (−0.794 − 1.79i)12-s + (0.226 + 0.845i)13-s + (0.280 − 0.587i)14-s + (−1.72 + 0.932i)15-s + (0.211 − 0.977i)16-s + (0.00279 − 0.0104i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.996 - 0.0862i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.996 - 0.0862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0260188 + 0.602005i\)
\(L(\frac12)\) \(\approx\) \(0.0260188 + 0.602005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.470i)T \)
5 \( 1 + (1.53 + 1.62i)T \)
19 \( 1 + (-3.17 + 2.98i)T \)
good3 \( 1 + (-0.879 + 3.28i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.21 - 1.21i)T - 7iT^{2} \)
11 \( 1 + 2.37iT - 11T^{2} \)
13 \( 1 + (-0.816 - 3.04i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.0115 + 0.0430i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (3.19 - 0.857i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.14 - 2.39i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.08iT - 31T^{2} \)
37 \( 1 + (-1.11 - 1.11i)T + 37iT^{2} \)
41 \( 1 + (4.95 + 8.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.25 + 4.66i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.09 + 4.09i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.09 + 7.81i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (7.40 + 12.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.96 - 6.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0397 + 0.148i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.923 + 0.533i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.0287 - 0.00770i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.12 - 1.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.41 - 2.41i)T + 83iT^{2} \)
89 \( 1 + (0.155 + 0.0894i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.93 + 10.9i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.22860520737384328151196950121, −9.399596928173644675528431467707, −8.742801898660484091952078414269, −8.148343037487158344204611078609, −7.26327028550210309538823182323, −6.50162404492681779012602709143, −5.55016499914139517290648536686, −3.19014789879144986663607785687, −1.83770393051102254542198286697, −0.49857177944703565805652804982, 2.84822498522227346372614603919, 3.51120153861646813885310626175, 4.51384623225561592297486247531, 6.17169073460534192065214028669, 7.64451426352718351021111632058, 8.247752898137590718873585911163, 9.385420234874435917788900330701, 10.16790740454723707674075970105, 10.43942835298322908961546517975, 11.35559209850301583238452859219

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