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Properties

Label 2-1380-115.22-c1-0-20
Degree $2$
Conductor $1380$
Sign $0.398 + 0.917i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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  + (0.707 − 0.707i)3-s + (1.98 − 1.02i)5-s + (1.54 − 1.54i)7-s − 1.00i·9-s + 1.73i·11-s + (2.14 − 2.14i)13-s + (0.676 − 2.13i)15-s + (0.622 − 0.622i)17-s − 1.52·19-s − 2.17i·21-s + (−0.781 − 4.73i)23-s + (2.88 − 4.08i)25-s + (−0.707 − 0.707i)27-s + 8.11i·29-s + 3.88·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.887 − 0.460i)5-s + (0.582 − 0.582i)7-s − 0.333i·9-s + 0.524i·11-s + (0.595 − 0.595i)13-s + (0.174 − 0.550i)15-s + (0.150 − 0.150i)17-s − 0.348·19-s − 0.475i·21-s + (−0.162 − 0.986i)23-s + (0.576 − 0.817i)25-s + (−0.136 − 0.136i)27-s + 1.50i·29-s + 0.697·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.398 + 0.917i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.398 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.453856466\)
\(L(\frac12)\) \(\approx\) \(2.453856466\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-1.98 + 1.02i)T \)
23 \( 1 + (0.781 + 4.73i)T \)
good7 \( 1 + (-1.54 + 1.54i)T - 7iT^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (-2.14 + 2.14i)T - 13iT^{2} \)
17 \( 1 + (-0.622 + 0.622i)T - 17iT^{2} \)
19 \( 1 + 1.52T + 19T^{2} \)
29 \( 1 - 8.11iT - 29T^{2} \)
31 \( 1 - 3.88T + 31T^{2} \)
37 \( 1 + (4.35 - 4.35i)T - 37iT^{2} \)
41 \( 1 - 5.64T + 41T^{2} \)
43 \( 1 + (5.05 + 5.05i)T + 43iT^{2} \)
47 \( 1 + (-2.65 - 2.65i)T + 47iT^{2} \)
53 \( 1 + (6.53 + 6.53i)T + 53iT^{2} \)
59 \( 1 - 1.33iT - 59T^{2} \)
61 \( 1 + 3.35iT - 61T^{2} \)
67 \( 1 + (0.825 - 0.825i)T - 67iT^{2} \)
71 \( 1 + 9.10T + 71T^{2} \)
73 \( 1 + (-7.83 + 7.83i)T - 73iT^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \)
89 \( 1 - 3.23T + 89T^{2} \)
97 \( 1 + (1.44 - 1.44i)T - 97iT^{2} \)
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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

−9.323377304957695236083436249219, −8.574302144154534265091899379320, −7.956937575167169535188558420290, −6.96399490753783181065578938653, −6.23789554083390580533188392107, −5.18201081428548984171717555218, −4.42914909692149413266431766042, −3.16371170761076407528153545905, −1.99175236835880074519492832825, −1.04644972172181346311023556841, 1.62534511348777311717955100047, 2.54595652426837612749041258133, 3.60331839917425566281087469530, 4.65551865357067639206344613465, 5.73037702565589345747345092454, 6.20098137699547150639729899880, 7.37488175565640704914225442628, 8.307187447594271292038461571487, 8.961787429952607584213570191742, 9.676314606066805723023312181323

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