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Properties

Label 2-1300-13.10-c1-0-21
Degree $2$
Conductor $1300$
Sign $-0.982 - 0.184i$
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  + (−1.16 − 2.01i)3-s + (−0.346 − 0.199i)7-s + (−1.21 + 2.11i)9-s + (1.5 − 0.866i)11-s + (0.619 − 3.55i)13-s + (−0.346 + 0.599i)17-s + (−4.65 − 2.68i)19-s + 0.932i·21-s + (0.0535 + 0.0927i)23-s − 1.30·27-s + (2.45 + 4.24i)29-s − 7.86i·31-s + (−3.49 − 2.01i)33-s + (−1.96 + 1.13i)37-s + (−7.89 + 2.89i)39-s + ⋯
L(s)  = 1  + (−0.673 − 1.16i)3-s + (−0.130 − 0.0755i)7-s + (−0.406 + 0.704i)9-s + (0.452 − 0.261i)11-s + (0.171 − 0.985i)13-s + (−0.0839 + 0.145i)17-s + (−1.06 − 0.616i)19-s + 0.203i·21-s + (0.0111 + 0.0193i)23-s − 0.251·27-s + (0.455 + 0.788i)29-s − 1.41i·31-s + (−0.608 − 0.351i)33-s + (−0.322 + 0.186i)37-s + (−1.26 + 0.462i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.982 - 0.184i$
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: \((2,\ 1300,\ (\ :1/2),\ -0.982 - 0.184i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6909360533\)
\(L(\frac12)\) \(\approx\) \(0.6909360533\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.619 + 3.55i)T \)
good3 \( 1 + (1.16 + 2.01i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.346 + 0.199i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.346 - 0.599i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.65 + 2.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0535 - 0.0927i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.45 - 4.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 + (1.96 - 1.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.69 + 3.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.00 - 5.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + (6.30 + 3.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.34 - 7.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.15 + 0.664i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.35 + 1.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 + (-0.300 + 0.173i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.66 - 4.42i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.119095595915456725048507569170, −8.236850459653244743660750834784, −7.51251651583699636816966809841, −6.60087340081595935399541686834, −6.14149825435916902768284096102, −5.21406837508971164090102105714, −4.02417205531513523622332877152, −2.76844918802942074130460817953, −1.50642197614219762074019598255, −0.31743406299741299603245568764, 1.76579568957088997854058890649, 3.30209102742426308396984491457, 4.36486708267451673987501848980, 4.71965871269135045768658557234, 5.98133885671716615549204568984, 6.49523901210332374020702691205, 7.64058812883793476114116980138, 8.767658076216356193403684579758, 9.360803385220767873182434702311, 10.13713012419395278213335348889

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