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Properties

Label 2-65-65.7-c1-0-3
Degree $2$
Conductor $65$
Sign $0.281 + 0.959i$
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  + (−0.427 − 0.246i)2-s + (−0.243 − 0.908i)3-s + (−0.878 − 1.52i)4-s + (−0.284 − 2.21i)5-s + (−0.120 + 0.448i)6-s + (1.83 + 3.18i)7-s + 1.85i·8-s + (1.83 − 1.05i)9-s + (−0.426 + 1.01i)10-s + (−0.177 − 0.664i)11-s + (−1.16 + 1.16i)12-s + (−2.92 + 2.11i)13-s − 1.81i·14-s + (−1.94 + 0.798i)15-s + (−1.29 + 2.24i)16-s + (2.29 + 0.614i)17-s + ⋯
L(s)  = 1  + (−0.302 − 0.174i)2-s + (−0.140 − 0.524i)3-s + (−0.439 − 0.760i)4-s + (−0.127 − 0.991i)5-s + (−0.0490 + 0.183i)6-s + (0.694 + 1.20i)7-s + 0.655i·8-s + (0.610 − 0.352i)9-s + (−0.134 + 0.322i)10-s + (−0.0536 − 0.200i)11-s + (−0.337 + 0.337i)12-s + (−0.810 + 0.585i)13-s − 0.485i·14-s + (−0.502 + 0.206i)15-s + (−0.324 + 0.562i)16-s + (0.556 + 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.281 + 0.959i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.582322 - 0.436185i\)
\(L(\frac12)\) \(\approx\) \(0.582322 - 0.436185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.284 + 2.21i)T \)
13 \( 1 + (2.92 - 2.11i)T \)
good2 \( 1 + (0.427 + 0.246i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.243 + 0.908i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.83 - 3.18i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.177 + 0.664i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.29 - 0.614i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.29 - 1.41i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.30 + 0.350i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (8.24 + 4.75i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.81 - 4.81i)T + 31iT^{2} \)
37 \( 1 + (0.917 - 1.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.534 - 0.143i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.560 - 2.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 3.80T + 47T^{2} \)
53 \( 1 + (2.47 - 2.47i)T - 53iT^{2} \)
59 \( 1 + (2.69 - 10.0i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.09 + 5.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.6 + 6.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.73 - 6.47i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 3.37iT - 73T^{2} \)
79 \( 1 - 3.12iT - 79T^{2} \)
83 \( 1 - 2.13T + 83T^{2} \)
89 \( 1 + (-3.26 + 0.874i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-6.12 + 3.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.70923772711220539872615408647, −13.51587895856440888201125608648, −12.25906308352914227387036727239, −11.62453130695327603604065035541, −9.807832172286613526455916638719, −9.009175614638137520895738359459, −7.78749380709282734997390476197, −5.82239224600822766260727798930, −4.78967095557496256949660213614, −1.58077064638148416848972069651, 3.51417157787690166020860247077, 4.84683710638821427977951804903, 7.37566146090300836140953140618, 7.59292602498354729656286397160, 9.628496737169697969486661669173, 10.44599434715628388856834479832, 11.57666662090569415875560760990, 13.07731063234616716268420672656, 14.08758352591801375893395154295, 15.17111955510005159917186281021

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