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Properties

Label 2-2380-2380.1427-c0-0-13
Degree $2$
Conductor $2380$
Sign $0.973 + 0.229i$
Analytic cond. $1.18777$
Root an. cond. $1.08985$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (−0.831 − 0.831i)3-s + (0.309 + 0.951i)4-s + (0.891 − 0.453i)5-s + (−0.183 − 1.16i)6-s + (0.707 − 0.707i)7-s + (−0.309 + 0.951i)8-s + 0.381i·9-s + (0.987 + 0.156i)10-s + (0.533 − 1.04i)12-s + (0.987 − 0.156i)14-s + (−1.11 − 0.363i)15-s + (−0.809 + 0.587i)16-s + (0.707 + 0.707i)17-s + (−0.224 + 0.309i)18-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (−0.831 − 0.831i)3-s + (0.309 + 0.951i)4-s + (0.891 − 0.453i)5-s + (−0.183 − 1.16i)6-s + (0.707 − 0.707i)7-s + (−0.309 + 0.951i)8-s + 0.381i·9-s + (0.987 + 0.156i)10-s + (0.533 − 1.04i)12-s + (0.987 − 0.156i)14-s + (−1.11 − 0.363i)15-s + (−0.809 + 0.587i)16-s + (0.707 + 0.707i)17-s + (−0.224 + 0.309i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2380\)    =    \(2^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(1.18777\)
Root analytic conductor: \(1.08985\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2380} (1427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2380,\ (\ :0),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.804353955\)
\(L(\frac12)\) \(\approx\) \(1.804353955\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.891 + 0.453i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.831 + 0.831i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.78iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 1.97T + T^{2} \)
43 \( 1 + (-1.39 - 1.39i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 0.312T + T^{2} \)
67 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.831 + 0.831i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1.34 - 1.34i)T + iT^{2} \)
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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

−8.911891993377896675337411632557, −7.993662019268856114409766601589, −7.47631114810048374222526046124, −6.56650362363834154146532716905, −6.01738072901133072049136130171, −5.38420700329736902499584331936, −4.60529365911699218447118632023, −3.66256342478411470149834959669, −2.21646506624696078350597497152, −1.21768501701326816428583118927, 1.54230008198984253606168255388, 2.54958594560608865707222367412, 3.47854766258025457389204721776, 4.65878687093723449061978849411, 5.25804945946505641469536869079, 5.63271652249636834143633130997, 6.49342007248527301296758549064, 7.39901205875067714504442960594, 8.772325340130916460187937939525, 9.451909540742578956772541319365

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