↑

Properties

Label 2-1300-65.8-c1-0-11
Degree $2$
Conductor $1300$
Sign $0.910 + 0.414i$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.285 + 0.285i)3-s + 1.26i·7-s − 2.83i·9-s + (4.21 − 4.21i)11-s + (−2.28 + 2.78i)13-s + (1.93 + 1.93i)17-s + (−2.55 + 2.55i)19-s + (−0.361 + 0.361i)21-s + (5.21 − 5.21i)23-s + (1.66 − 1.66i)27-s − 7.16i·29-s + (2.78 + 2.78i)31-s + 2.40·33-s − 5.83i·37-s + (−1.44 + 0.143i)39-s + ⋯
L(s)  = 1  + (0.164 + 0.164i)3-s + 0.478i·7-s − 0.945i·9-s + (1.27 − 1.27i)11-s + (−0.633 + 0.773i)13-s + (0.468 + 0.468i)17-s + (−0.585 + 0.585i)19-s + (−0.0788 + 0.0788i)21-s + (1.08 − 1.08i)23-s + (0.320 − 0.320i)27-s − 1.33i·29-s + (0.500 + 0.500i)31-s + 0.419·33-s − 0.959i·37-s + (−0.231 + 0.0230i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.910 + 0.414i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.910 + 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.831657720\)
\(L(\frac12)\) \(\approx\) \(1.831657720\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 + (2.28 - 2.78i)T \)
good3 \( 1 + (-0.285 - 0.285i)T + 3iT^{2} \)
7 \( 1 - 1.26iT - 7T^{2} \)
11 \( 1 + (-4.21 + 4.21i)T - 11iT^{2} \)
17 \( 1 + (-1.93 - 1.93i)T + 17iT^{2} \)
19 \( 1 + (2.55 - 2.55i)T - 19iT^{2} \)
23 \( 1 + (-5.21 + 5.21i)T - 23iT^{2} \)
29 \( 1 + 7.16iT - 29T^{2} \)
31 \( 1 + (-2.78 - 2.78i)T + 31iT^{2} \)
37 \( 1 + 5.83iT - 37T^{2} \)
41 \( 1 + (1.33 + 1.33i)T + 41iT^{2} \)
43 \( 1 + (5.65 - 5.65i)T - 43iT^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + (-9.27 - 9.27i)T + 53iT^{2} \)
59 \( 1 + (2.55 + 2.55i)T + 59iT^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 1.30T + 67T^{2} \)
71 \( 1 + (-8.72 - 8.72i)T + 71iT^{2} \)
73 \( 1 - 7.64T + 73T^{2} \)
79 \( 1 - 0.954iT - 79T^{2} \)
83 \( 1 + 5.13iT - 83T^{2} \)
89 \( 1 + (-5.31 - 5.31i)T + 89iT^{2} \)
97 \( 1 + 2.81T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.454286808753820581156137485919, −8.777648918400569356106020991295, −8.313847892888867226635593783068, −6.87765908479374228675104590040, −6.37841185802139381107045131886, −5.53209201731786774333198010448, −4.21513879260971272032889116329, −3.58971840644114921528481819169, −2.39938558239425568353706246253, −0.893562878388828796298350770363, 1.24619679046627161696119522813, 2.42469826903994600865210417040, 3.58339777388987831551854573581, 4.73555786503731318263551787388, 5.25626321301036341615054023783, 6.76172426485937299196573222450, 7.17641369445507219051076373079, 7.959443423775840310935021132318, 8.957271064576117650927903115311, 9.744748140291073529895128347989

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