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Properties

Label 2-1300-65.2-c1-0-6
Degree $2$
Conductor $1300$
Sign $-0.00863 - 0.999i$
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  + (0.5 + 0.133i)3-s + (2.13 + 1.23i)7-s + (−2.36 − 1.36i)9-s + (−1.13 + 4.23i)11-s + (2 + 3i)13-s + (0.232 + 0.866i)17-s + (−2.86 + 0.767i)19-s + (0.901 + 0.901i)21-s + (0.0358 − 0.133i)23-s + (−2.09 − 2.09i)27-s + (−1.5 + 0.866i)29-s + (−5.19 + 5.19i)31-s + (−1.13 + 1.96i)33-s + (−1.33 + 0.767i)37-s + (0.598 + 1.76i)39-s + ⋯
L(s)  = 1  + (0.288 + 0.0773i)3-s + (0.806 + 0.465i)7-s + (−0.788 − 0.455i)9-s + (−0.341 + 1.27i)11-s + (0.554 + 0.832i)13-s + (0.0562 + 0.210i)17-s + (−0.657 + 0.176i)19-s + (0.196 + 0.196i)21-s + (0.00748 − 0.0279i)23-s + (−0.403 − 0.403i)27-s + (−0.278 + 0.160i)29-s + (−0.933 + 0.933i)31-s + (−0.197 + 0.341i)33-s + (−0.218 + 0.126i)37-s + (0.0957 + 0.283i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00863 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00863 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.00863 - 0.999i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ -0.00863 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.554244674\)
\(L(\frac12)\) \(\approx\) \(1.554244674\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 3i)T \)
good3 \( 1 + (-0.5 - 0.133i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-2.13 - 1.23i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.13 - 4.23i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.232 - 0.866i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.86 - 0.767i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.0358 + 0.133i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.5 - 0.866i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.19 - 5.19i)T - 31iT^{2} \)
37 \( 1 + (1.33 - 0.767i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.33 - 2.5i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.96 + 1.59i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 + (2.46 - 2.46i)T - 53iT^{2} \)
59 \( 1 + (-2.33 - 8.69i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.13 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.598 - 2.23i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 0.535iT - 79T^{2} \)
83 \( 1 - 2.92iT - 83T^{2} \)
89 \( 1 + (14.7 + 3.96i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.86 - 6.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.612015777144208192917367033618, −9.057062624507454144794704691101, −8.312053628092163734898431927184, −7.53552339801844861001235311117, −6.54441692001305465765941146047, −5.66556422304777833492521121597, −4.71121704520052090884210641254, −3.86347265416512004997542015074, −2.56806877926934140371085430867, −1.64874071714632276173618973925, 0.61568853857159655074151246325, 2.18378031236718945591933123760, 3.22067364091488973188222961943, 4.19025498037766928299365450632, 5.46463054955118823946635827560, 5.83752318110588018293643522136, 7.18482987284975876899492109511, 8.078487285527107267860835242439, 8.386791161574011823077736350357, 9.281418779162684832346601939521

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