Properties

Label 2-1380-345.344-c1-0-2
Degree $2$
Conductor $1380$
Sign $-0.635 + 0.772i$
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  + (0.779 + 1.54i)3-s + (−0.268 + 2.21i)5-s − 2.49·7-s + (−1.78 + 2.41i)9-s − 3.17·11-s − 5.31i·13-s + (−3.64 + 1.31i)15-s − 2.70i·17-s + 8.13i·19-s + (−1.94 − 3.86i)21-s + (4.12 − 2.44i)23-s + (−4.85 − 1.19i)25-s + (−5.12 − 0.877i)27-s − 2.82i·29-s − 6.28·31-s + ⋯
L(s)  = 1  + (0.450 + 0.892i)3-s + (−0.119 + 0.992i)5-s − 0.944·7-s + (−0.594 + 0.804i)9-s − 0.958·11-s − 1.47i·13-s + (−0.940 + 0.339i)15-s − 0.655i·17-s + 1.86i·19-s + (−0.425 − 0.843i)21-s + (0.859 − 0.510i)23-s + (−0.971 − 0.238i)25-s + (−0.985 − 0.168i)27-s − 0.524i·29-s − 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.635 + 0.772i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.635 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2925035970\)
\(L(\frac12)\) \(\approx\) \(0.2925035970\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.779 - 1.54i)T \)
5 \( 1 + (0.268 - 2.21i)T \)
23 \( 1 + (-4.12 + 2.44i)T \)
good7 \( 1 + 2.49T + 7T^{2} \)
11 \( 1 + 3.17T + 11T^{2} \)
13 \( 1 + 5.31iT - 13T^{2} \)
17 \( 1 + 2.70iT - 17T^{2} \)
19 \( 1 - 8.13iT - 19T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 6.28T + 31T^{2} \)
37 \( 1 + 3.82T + 37T^{2} \)
41 \( 1 - 3.17iT - 41T^{2} \)
43 \( 1 + 6.93T + 43T^{2} \)
47 \( 1 - 8.10T + 47T^{2} \)
53 \( 1 + 3.79iT - 53T^{2} \)
59 \( 1 + 11.2iT - 59T^{2} \)
61 \( 1 + 0.443iT - 61T^{2} \)
67 \( 1 - 6.68T + 67T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 - 2.80iT - 73T^{2} \)
79 \( 1 + 6.08iT - 79T^{2} \)
83 \( 1 - 8.29iT - 83T^{2} \)
89 \( 1 + 9.80T + 89T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
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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

−10.05865160300038341479475768529, −9.638753333450773605928383841003, −8.362592059751302405563299865635, −7.83840105140968961550995104891, −6.91279415708917096443901510605, −5.79998433083142432141830971853, −5.20659026449368894391799628421, −3.76440030800466570181225262512, −3.20238185428031203144816223096, −2.45566419179251218904561866355, 0.10558920265300754765282991566, 1.56590022511011262054751907341, 2.68142484772255747122694730687, 3.74296059983746331345598231127, 4.87559489102517709726185863285, 5.79269430690354729721500322237, 6.88157213070749237800864614875, 7.25959926222661875190193579964, 8.400775342419632034668600194463, 9.106275609209627507362186937338

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