Properties

Label 2-1300-65.4-c1-0-13
Degree $2$
Conductor $1300$
Sign $-0.0518 + 0.998i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0518 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0518 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.0518 + 0.998i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ -0.0518 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.199975288\)
\(L(\frac12)\) \(\approx\) \(1.199975288\)
\(L(\frac{3}{2})\) 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

−9.566271986091752948413494492333, −8.537290670529568344696286600930, −7.991321867086713439525341187555, −6.85805423640665175626805128431, −6.14648906865127404102269222251, −5.30748279498656951529297945236, −4.24075595123390294780679591691, −3.24321044062906512530647808284, −2.15088238698401561511076746246, −0.50638658977768569345835081980, 1.48623815818085756777760600428, 2.66599509450046246079130772969, 3.85320743139254781034987795296, 4.82046786858116777570410152792, 5.57392121074333715764764597482, 6.72257781495767999862709820824, 7.30013590552317735577422411352, 8.415285450516491757017551129317, 8.877001061375964416309128448870, 9.960917081272048779025924318546

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