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Properties

Label 2-1300-1300.947-c0-0-0
Degree $2$
Conductor $1300$
Sign $0.331 - 0.943i$
Analytic cond. $0.648784$
Root an. cond. $0.805471$
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  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (−0.104 − 0.994i)5-s + (−0.587 − 0.809i)8-s + (−0.207 + 0.978i)9-s + (0.951 − 0.309i)10-s + (0.978 − 0.207i)13-s + (0.669 − 0.743i)16-s + (1.55 + 1.25i)17-s − 1.00·18-s + (0.5 + 0.866i)20-s + (−0.978 + 0.207i)25-s + (0.406 + 0.913i)26-s + (1.94 + 0.204i)29-s + (0.866 + 0.500i)32-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (−0.104 − 0.994i)5-s + (−0.587 − 0.809i)8-s + (−0.207 + 0.978i)9-s + (0.951 − 0.309i)10-s + (0.978 − 0.207i)13-s + (0.669 − 0.743i)16-s + (1.55 + 1.25i)17-s − 1.00·18-s + (0.5 + 0.866i)20-s + (−0.978 + 0.207i)25-s + (0.406 + 0.913i)26-s + (1.94 + 0.204i)29-s + (0.866 + 0.500i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.331 - 0.943i$
Analytic conductor: \(0.648784\)
Root analytic conductor: \(0.805471\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (947, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :0),\ 0.331 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.101631054\)
\(L(\frac12)\) \(\approx\) \(1.101631054\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.207 - 0.978i)T \)
5 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (-0.978 + 0.207i)T \)
good3 \( 1 + (0.207 - 0.978i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.406 + 0.913i)T^{2} \)
17 \( 1 + (-1.55 - 1.25i)T + (0.207 + 0.978i)T^{2} \)
19 \( 1 + (-0.207 - 0.978i)T^{2} \)
23 \( 1 + (-0.406 - 0.913i)T^{2} \)
29 \( 1 + (-1.94 - 0.204i)T + (0.978 + 0.207i)T^{2} \)
31 \( 1 + (-0.951 - 0.309i)T^{2} \)
37 \( 1 + (0.278 - 0.309i)T + (-0.104 - 0.994i)T^{2} \)
41 \( 1 + (1.25 - 0.0658i)T + (0.994 - 0.104i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-1.84 + 0.292i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.406 - 0.913i)T^{2} \)
61 \( 1 + (0.544 + 0.604i)T + (-0.104 + 0.994i)T^{2} \)
67 \( 1 + (0.669 + 0.743i)T^{2} \)
71 \( 1 + (-0.743 - 0.669i)T^{2} \)
73 \( 1 + (1.73 + 0.564i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.761 + 0.494i)T + (0.406 + 0.913i)T^{2} \)
97 \( 1 + (-0.773 - 1.73i)T + (-0.669 + 0.743i)T^{2} \)
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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

−9.966616200711790497127465902264, −8.721843163237228389142110752879, −8.326680599978956667516363804746, −7.81488292152008758616419602814, −6.63651764779722009120146543321, −5.70087810167328362856055954718, −5.16604570460652871889151989588, −4.22170727777200558131576793732, −3.26920777491487757376289322464, −1.35600053787039826591889136651, 1.13743336802455032979532485401, 2.78149738123718024268662046922, 3.31318196718020695759776581546, 4.23455438214126710672625581974, 5.48185665150740415863130180658, 6.25901690733330297874284236692, 7.16964146951134419150066099559, 8.288839971200459252151156637362, 9.067583351606407766484392563734, 9.987136531766691494282754590759

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