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Properties

Label 2-65-13.12-c1-0-4
Degree $2$
Conductor $65$
Sign $-0.223 + 0.974i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
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  − 2.51i·2-s + 1.51·3-s − 4.32·4-s + i·5-s − 3.80i·6-s + 3.32i·7-s + 5.83i·8-s − 0.707·9-s + 2.51·10-s − 2.83i·11-s − 6.54·12-s + (−3.51 − 0.806i)13-s + 8.34·14-s + 1.51i·15-s + 6.02·16-s + 6.64·17-s + ⋯
L(s)  = 1  − 1.77i·2-s + 0.874·3-s − 2.16·4-s + 0.447i·5-s − 1.55i·6-s + 1.25i·7-s + 2.06i·8-s − 0.235·9-s + 0.795·10-s − 0.854i·11-s − 1.88·12-s + (−0.974 − 0.223i)13-s + 2.23·14-s + 0.390i·15-s + 1.50·16-s + 1.61·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.223 + 0.974i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.223 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613037 - 0.769721i\)
\(L(\frac12)\) \(\approx\) \(0.613037 - 0.769721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
13 \( 1 + (3.51 + 0.806i)T \)
good2 \( 1 + 2.51iT - 2T^{2} \)
3 \( 1 - 1.51T + 3T^{2} \)
7 \( 1 - 3.32iT - 7T^{2} \)
11 \( 1 + 2.83iT - 11T^{2} \)
17 \( 1 - 6.64T + 17T^{2} \)
19 \( 1 + 2.19iT - 19T^{2} \)
23 \( 1 - 0.485T + 23T^{2} \)
29 \( 1 + 3.32T + 29T^{2} \)
31 \( 1 + 3.80iT - 31T^{2} \)
37 \( 1 - 9.32iT - 37T^{2} \)
41 \( 1 - 1.61iT - 41T^{2} \)
43 \( 1 - 0.872T + 43T^{2} \)
47 \( 1 + 3.32iT - 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 8.83iT - 59T^{2} \)
61 \( 1 + 3.70T + 61T^{2} \)
67 \( 1 + 4.29iT - 67T^{2} \)
71 \( 1 + 2.19iT - 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 - 0.585T + 79T^{2} \)
83 \( 1 + 7.70iT - 83T^{2} \)
89 \( 1 - 3.41iT - 89T^{2} \)
97 \( 1 + 0.641iT - 97T^{2} \)
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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

−14.32786399171178869870163739693, −13.30551433024454737753302607472, −12.12210810935846824603059680982, −11.41651577707671778762188238346, −10.02664614708870365503595157855, −9.137658412270706370707550165582, −8.099435753844659029482053967235, −5.44998258684393341157446975988, −3.32437052564799633022118231743, −2.48887610981466579988293126407, 4.04651105794086478505517210749, 5.46288159275398275192089909136, 7.27787089118481219684821541918, 7.75043303720510673533145671322, 9.068382569454427754361420392908, 10.06075000967323738220455540702, 12.42003690771437732023442424975, 13.67014887039742368222812187071, 14.39820541552617185893150368689, 14.97177182291491666067663068449

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