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Properties

Label 2-380-76.27-c1-0-33
Degree $2$
Conductor $380$
Sign $-0.955 + 0.293i$
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  + (−0.641 − 1.26i)2-s + (1.09 − 1.89i)3-s + (−1.17 + 1.61i)4-s + (0.5 − 0.866i)5-s + (−3.09 − 0.163i)6-s − 2.65i·7-s + (2.79 + 0.445i)8-s + (−0.901 − 1.56i)9-s + (−1.41 − 0.0746i)10-s − 1.44i·11-s + (1.77 + 4.00i)12-s + (1.00 − 0.577i)13-s + (−3.34 + 1.70i)14-s + (−1.09 − 1.89i)15-s + (−1.22 − 3.80i)16-s + (−1.39 + 2.42i)17-s + ⋯
L(s)  = 1  + (−0.453 − 0.891i)2-s + (0.632 − 1.09i)3-s + (−0.588 + 0.808i)4-s + (0.223 − 0.387i)5-s + (−1.26 − 0.0667i)6-s − 1.00i·7-s + (0.987 + 0.157i)8-s + (−0.300 − 0.520i)9-s + (−0.446 − 0.0235i)10-s − 0.435i·11-s + (0.513 + 1.15i)12-s + (0.277 − 0.160i)13-s + (−0.893 + 0.454i)14-s + (−0.282 − 0.490i)15-s + (−0.307 − 0.951i)16-s + (−0.339 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.955 + 0.293i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.955 + 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.184444 - 1.22780i\)
\(L(\frac12)\) \(\approx\) \(0.184444 - 1.22780i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.641 + 1.26i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (4.25 + 0.941i)T \)
good3 \( 1 + (-1.09 + 1.89i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 2.65iT - 7T^{2} \)
11 \( 1 + 1.44iT - 11T^{2} \)
13 \( 1 + (-1.00 + 0.577i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.39 - 2.42i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.05 + 1.18i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.87 - 2.81i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.90T + 31T^{2} \)
37 \( 1 - 7.08iT - 37T^{2} \)
41 \( 1 + (8.49 + 4.90i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.24 - 3.02i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.81 - 1.04i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.42 + 3.70i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.64 + 2.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.74 - 11.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.75 + 9.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.58 - 2.75i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.83 + 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.51 + 9.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.78iT - 83T^{2} \)
89 \( 1 + (-1.26 + 0.727i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.0 - 5.80i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.82072311781604100914571585326, −10.20465699605147810588468370467, −8.867051706052061216516088975466, −8.351725783048101695535290932061, −7.44763868834130691732978219696, −6.48364153987366213202307986879, −4.67683319657282440651915512790, −3.47096365326958445617884748312, −2.13197768018859585424435039365, −0.961103078807122524755373370687, 2.35353560550729298866436624682, 3.94532645317995034749727707128, 5.00192300669226319365379238948, 6.05211636346080040292626487741, 7.08381588343505240495864334509, 8.365135126323408452851943125674, 9.004789295704678928511474314492, 9.699413344598328703409826982878, 10.42837390152053533434616247503, 11.48230564992273021644087851357

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