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Properties

Label 2-65-13.7-c2-0-6
Degree $2$
Conductor $65$
Sign $-0.370 + 0.928i$
Analytic cond. $1.77112$
Root an. cond. $1.33083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.23 + 0.598i)2-s + (2.03 − 3.51i)3-s + (1.17 − 0.676i)4-s + (−1.58 − 1.58i)5-s + (−2.43 + 9.07i)6-s + (−9.58 − 2.56i)7-s + (4.32 − 4.32i)8-s + (−3.74 − 6.49i)9-s + (4.48 + 2.58i)10-s + (−2.41 − 9.02i)11-s − 5.49i·12-s + (1.86 − 12.8i)13-s + 22.9·14-s + (−8.77 + 2.35i)15-s + (−9.79 + 16.9i)16-s + (13.7 − 7.95i)17-s + ⋯
L(s)  = 1  + (−1.11 + 0.299i)2-s + (0.676 − 1.17i)3-s + (0.293 − 0.169i)4-s + (−0.316 − 0.316i)5-s + (−0.405 + 1.51i)6-s + (−1.36 − 0.366i)7-s + (0.541 − 0.541i)8-s + (−0.416 − 0.721i)9-s + (0.448 + 0.258i)10-s + (−0.219 − 0.820i)11-s − 0.458i·12-s + (0.143 − 0.989i)13-s + 1.63·14-s + (−0.584 + 0.156i)15-s + (−0.611 + 1.05i)16-s + (0.810 − 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.370 + 0.928i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ -0.370 + 0.928i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.336157 - 0.496240i\)
\(L(\frac12)\) \(\approx\) \(0.336157 - 0.496240i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.58 + 1.58i)T \)
13 \( 1 + (-1.86 + 12.8i)T \)
good2 \( 1 + (2.23 - 0.598i)T + (3.46 - 2i)T^{2} \)
3 \( 1 + (-2.03 + 3.51i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + (9.58 + 2.56i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (2.41 + 9.02i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (-13.7 + 7.95i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (6.92 - 25.8i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-13.8 - 7.99i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-16.4 + 28.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-11.4 - 11.4i)T + 961iT^{2} \)
37 \( 1 + (8.33 + 31.1i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-66.9 + 17.9i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (12.7 - 7.35i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (16.7 - 16.7i)T - 2.20e3iT^{2} \)
53 \( 1 + 87.0T + 2.80e3T^{2} \)
59 \( 1 + (-14.9 - 4.01i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (14.4 + 25.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (8.75 - 2.34i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-1.21 + 4.51i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-95.0 + 95.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 14.6T + 6.24e3T^{2} \)
83 \( 1 + (97.0 + 97.0i)T + 6.88e3iT^{2} \)
89 \( 1 + (-13.8 - 51.8i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (44.7 - 166. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−14.05654705005247407800066316292, −13.09525807068285562473111091134, −12.48317154673508608064287586562, −10.49632082153455970142454334947, −9.385579808908409737699397912949, −8.158002815452085742441160406582, −7.58569142844181326221227831870, −6.25071986730361713120640337197, −3.33285447681517355050808359705, −0.75036883202093965404329256098, 2.89800319820324276788594820440, 4.55862037735586975001808776581, 6.86688381173384297478104549596, 8.536269150293006288631545130784, 9.441761679393429975573636704385, 9.984889825752162868997026060355, 11.08184567105381307398975472886, 12.67260372672015868011954603247, 14.17188899065767780337180094522, 15.23199860965745234536877560205

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