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Properties

Label 2-1300-13.10-c1-0-20
Degree $2$
Conductor $1300$
Sign $-0.400 + 0.916i$
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  + (−0.0473 − 0.0820i)3-s + (−0.716 − 0.413i)7-s + (1.49 − 2.59i)9-s + (1.5 − 0.866i)11-s + (−3.32 − 1.40i)13-s + (−0.716 + 1.24i)17-s + (−0.926 − 0.534i)19-s + 0.0783i·21-s + (−1.54 − 2.67i)23-s − 0.567·27-s + (−3.72 − 6.45i)29-s + 5.84i·31-s + (−0.142 − 0.0820i)33-s + (−0.851 + 0.491i)37-s + (0.0424 + 0.339i)39-s + ⋯
L(s)  = 1  + (−0.0273 − 0.0473i)3-s + (−0.270 − 0.156i)7-s + (0.498 − 0.863i)9-s + (0.452 − 0.261i)11-s + (−0.921 − 0.388i)13-s + (−0.173 + 0.300i)17-s + (−0.212 − 0.122i)19-s + 0.0171i·21-s + (−0.321 − 0.557i)23-s − 0.109·27-s + (−0.692 − 1.19i)29-s + 1.04i·31-s + (−0.0247 − 0.0142i)33-s + (−0.140 + 0.0808i)37-s + (0.00680 + 0.0543i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.400 + 0.916i$
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: \((2,\ 1300,\ (\ :1/2),\ -0.400 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.095359902\)
\(L(\frac12)\) \(\approx\) \(1.095359902\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.32 + 1.40i)T \)
good3 \( 1 + (0.0473 + 0.0820i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.716 + 0.413i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.716 - 1.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.926 + 0.534i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.54 + 2.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.72 + 6.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.84iT - 31T^{2} \)
37 \( 1 + (0.851 - 0.491i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.69 - 2.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.77 + 8.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 0.334T + 53T^{2} \)
59 \( 1 + (9.98 + 5.76i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.35 - 2.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.9 + 6.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.46 - 4.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 + 0.252T + 79T^{2} \)
83 \( 1 + 5.67iT - 83T^{2} \)
89 \( 1 + (-3.98 + 2.29i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.25 + 4.76i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.475183912831573500021472595827, −8.657860367666470052660205338571, −7.72526891959741907194490135271, −6.82451092384296336604064806075, −6.25725389492201393930670499444, −5.15264509535211254437231172086, −4.12263787610609757396642267005, −3.30831731499593132993597689697, −1.98835664872604114856908539671, −0.44619269988721533031297274031, 1.63688028307500557084038409289, 2.66774102065867158006319620689, 3.98397187182884531953484054576, 4.78611894188986557329392902062, 5.67147026001342185927213227644, 6.75688222228255308624469638912, 7.42522677545455498488380694360, 8.189149701983571041325834773594, 9.446759459148467735885291800247, 9.612925618110764636962131192155

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