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Properties

Label 2-650-13.12-c1-0-11
Degree $2$
Conductor $650$
Sign $0.277 + 0.960i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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  − i·2-s − 0.792·3-s − 4-s + 0.792i·6-s + 3.37i·7-s + i·8-s − 2.37·9-s − 5.04i·11-s + 0.792·12-s + (3.46 − i)13-s + 3.37·14-s + 16-s + 2.67·17-s + 2.37i·18-s − 3.46i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.457·3-s − 0.5·4-s + 0.323i·6-s + 1.27i·7-s + 0.353i·8-s − 0.790·9-s − 1.52i·11-s + 0.228·12-s + (0.960 − 0.277i)13-s + 0.901·14-s + 0.250·16-s + 0.648·17-s + 0.559i·18-s − 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913440 - 0.687051i\)
\(L(\frac12)\) \(\approx\) \(0.913440 - 0.687051i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
13 \( 1 + (-3.46 + i)T \)
good3 \( 1 + 0.792T + 3T^{2} \)
7 \( 1 - 3.37iT - 7T^{2} \)
11 \( 1 + 5.04iT - 11T^{2} \)
17 \( 1 - 2.67T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 - 6.63T + 23T^{2} \)
29 \( 1 - 8.74T + 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 8.11iT - 37T^{2} \)
41 \( 1 - 3.16iT - 41T^{2} \)
43 \( 1 + 9.30T + 43T^{2} \)
47 \( 1 - 4.62iT - 47T^{2} \)
53 \( 1 + 1.58T + 53T^{2} \)
59 \( 1 + 6.63iT - 59T^{2} \)
61 \( 1 - 4.74T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 3.96iT - 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 6.74T + 79T^{2} \)
83 \( 1 + 2.74iT - 83T^{2} \)
89 \( 1 - 8.51iT - 89T^{2} \)
97 \( 1 - 4.74iT - 97T^{2} \)
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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

−10.77877583163360881606139858195, −9.438481172149036491630863196447, −8.644881395226857282511668025127, −8.253097225701378565694029719323, −6.45397155392459845474812449839, −5.71662104221182252808114854055, −5.03444993858947179740143896978, −3.35323818360863410924567438543, −2.69795523700365249967317848351, −0.818752544098216797902701403214, 1.19902176720566936079191421228, 3.31849400409358216499858310009, 4.47077912402928445243419148596, 5.24819646503983321385143085266, 6.49593682682467919443343600681, 6.99549521282887535850288881952, 8.005612091632179693357235037305, 8.825030964559869005380039175784, 10.07072919194082727455029590753, 10.45967099472044281563684282099

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