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Properties

Label 2-1305-5.4-c1-0-56
Degree $2$
Conductor $1305$
Sign $0.306 + 0.951i$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
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  + 1.73i·2-s − 0.999·4-s + (−0.686 − 2.12i)5-s − 5.04i·7-s + 1.73i·8-s + (3.68 − 1.18i)10-s − 0.627·11-s + 4.25i·13-s + 8.74·14-s − 5·16-s − 1.58i·17-s − 4·19-s + (0.686 + 2.12i)20-s − 1.08i·22-s − 3.46i·23-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.499·4-s + (−0.306 − 0.951i)5-s − 1.90i·7-s + 0.612i·8-s + (1.16 − 0.375i)10-s − 0.189·11-s + 1.18i·13-s + 2.33·14-s − 1.25·16-s − 0.384i·17-s − 0.917·19-s + (0.153 + 0.475i)20-s − 0.231i·22-s − 0.722i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.306 + 0.951i$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ 0.306 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8579162563\)
\(L(\frac12)\) \(\approx\) \(0.8579162563\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.686 + 2.12i)T \)
29 \( 1 + T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
7 \( 1 + 5.04iT - 7T^{2} \)
11 \( 1 + 0.627T + 11T^{2} \)
13 \( 1 - 4.25iT - 13T^{2} \)
17 \( 1 + 1.58iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
31 \( 1 + 3.37T + 31T^{2} \)
37 \( 1 + 3.16iT - 37T^{2} \)
41 \( 1 - 4.74T + 41T^{2} \)
43 \( 1 + 10.8iT - 43T^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + 4.25iT - 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 1.87iT - 67T^{2} \)
71 \( 1 + 6.74T + 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 9.80iT - 83T^{2} \)
89 \( 1 + 0.744T + 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−9.136934184890405325557734012393, −8.518224144674297052768167019294, −7.61774571367510763131156807978, −7.11601202176170365369816251368, −6.43410180247871852000312606233, −5.27452693295993777412933709381, −4.43631103149416162867149898285, −3.89042212788785104759087022990, −1.91202766230064766523254954016, −0.33356172391887945225620828877, 1.77986386512124333407816739865, 2.79818578297069133206855960314, 3.13008792036218194031245019249, 4.43958314593660061371970823524, 5.76411745561887244511703311168, 6.26098378890219883023573557553, 7.51473067449316117400311525708, 8.336539956526798640929930572716, 9.259404528673911481359294642772, 9.950005961606170664285416234819

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