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Properties

Label 2-1300-65.49-c1-0-7
Degree $2$
Conductor $1300$
Sign $0.767 - 0.640i$
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  + (0.0820 − 0.0473i)3-s + (−0.413 + 0.716i)7-s + (−1.49 + 2.59i)9-s + (1.5 − 0.866i)11-s + (1.40 − 3.32i)13-s + (1.24 + 0.716i)17-s + (0.926 + 0.534i)19-s + 0.0783i·21-s + (2.67 − 1.54i)23-s + 0.567i·27-s + (3.72 + 6.45i)29-s + 5.84i·31-s + (0.0820 − 0.142i)33-s + (0.491 + 0.851i)37-s + (−0.0424 − 0.339i)39-s + ⋯
L(s)  = 1  + (0.0473 − 0.0273i)3-s + (−0.156 + 0.270i)7-s + (−0.498 + 0.863i)9-s + (0.452 − 0.261i)11-s + (0.388 − 0.921i)13-s + (0.300 + 0.173i)17-s + (0.212 + 0.122i)19-s + 0.0171i·21-s + (0.557 − 0.321i)23-s + 0.109i·27-s + (0.692 + 1.19i)29-s + 1.04i·31-s + (0.0142 − 0.0247i)33-s + (0.0808 + 0.140i)37-s + (−0.00680 − 0.0543i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.767 - 0.640i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.767 - 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.619322282\)
\(L(\frac12)\) \(\approx\) \(1.619322282\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 3.32i)T \)
good3 \( 1 + (-0.0820 + 0.0473i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.413 - 0.716i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.24 - 0.716i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.926 - 0.534i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.67 + 1.54i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.72 - 6.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.84iT - 31T^{2} \)
37 \( 1 + (-0.491 - 0.851i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.69 - 2.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.26 - 4.77i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 0.334iT - 53T^{2} \)
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 \( 1 + (6.87 + 11.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.46 - 4.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 - 0.252T + 79T^{2} \)
83 \( 1 - 5.67T + 83T^{2} \)
89 \( 1 + (3.98 - 2.29i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.76 - 8.25i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.744075887177113412918118628175, −8.748759631860220408121760571221, −8.277345676175211341699850982899, −7.35686846527299795609179914636, −6.37435970517781058940302779574, −5.52857076706821188426899120900, −4.79346983744484564001226437197, −3.44497050058262409016696462640, −2.68467279022884841184847076052, −1.19833562203825573771158789316, 0.798565267865760262825232877817, 2.29443525023836464638073113049, 3.55739724683567893273355330441, 4.22097461475760230223807816297, 5.43500781871466510663850629482, 6.37084976121951721976440428830, 6.96237000732513228422743158581, 7.969762735931916223295556277759, 8.931320144081069598462589299215, 9.453998971448917786876216882827

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