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Properties

Label 2-1300-13.10-c1-0-17
Degree $2$
Conductor $1300$
Sign $0.877 + 0.480i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.104 + 0.180i)3-s + (2.99 + 1.72i)7-s + (1.47 − 2.56i)9-s + (0.625 − 0.360i)11-s + (1.69 − 3.18i)13-s + (1.79 − 3.10i)17-s + (−6.51 − 3.76i)19-s + 0.721i·21-s + (−1.18 − 2.05i)23-s + 1.24·27-s + (3.68 + 6.37i)29-s + 0.668i·31-s + (0.130 + 0.0753i)33-s + (5.82 − 3.36i)37-s + (0.752 − 0.0267i)39-s + ⋯
L(s)  = 1  + (0.0602 + 0.104i)3-s + (1.13 + 0.653i)7-s + (0.492 − 0.853i)9-s + (0.188 − 0.108i)11-s + (0.468 − 0.883i)13-s + (0.435 − 0.754i)17-s + (−1.49 − 0.862i)19-s + 0.157i·21-s + (−0.247 − 0.428i)23-s + 0.239·27-s + (0.683 + 1.18i)29-s + 0.120i·31-s + (0.0227 + 0.0131i)33-s + (0.958 − 0.553i)37-s + (0.120 − 0.00428i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.877 + 0.480i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.877 + 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.006724519\)
\(L(\frac12)\) \(\approx\) \(2.006724519\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 + 3.18i)T \)
good3 \( 1 + (-0.104 - 0.180i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.99 - 1.72i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.625 + 0.360i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.79 + 3.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.51 + 3.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.18 + 2.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.68 - 6.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.668iT - 31T^{2} \)
37 \( 1 + (-5.82 + 3.36i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.32 - 3.65i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.07 - 7.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.40iT - 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + (-4.20 - 2.42i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.68 + 4.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.5 + 7.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.10 + 2.37i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.36iT - 73T^{2} \)
79 \( 1 + 5.20T + 79T^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 + (4.20 - 2.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.17 + 1.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.529462140305526165766562250043, −8.585447353622303494283182069180, −8.278550206952479297873686262259, −7.06406695555014582348872746812, −6.31718672534117730829140849989, −5.26938378059534021543583059909, −4.55493507734494642749648487267, −3.43921581933476980742258238729, −2.30738963928575364938098820691, −0.951424107698510533354228177396, 1.42078318263928718624518521260, 2.16431458985454936321704724211, 4.03666526372944743815323210023, 4.31236224796296195215718945040, 5.49837102747576540269967131893, 6.49499058882070968137803332098, 7.36163326891256065259434572646, 8.214187265170319086474281234713, 8.553899772452753857962852512899, 10.02046120702332494709726223372

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