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Properties

Label 2-65-5.4-c3-0-0
Degree $2$
Conductor $65$
Sign $0.134 - 0.990i$
Analytic cond. $3.83512$
Root an. cond. $1.95834$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.36i·2-s + 6.45i·3-s − 20.7·4-s + (−11.0 − 1.50i)5-s + 34.6·6-s + 12.0i·7-s + 68.6i·8-s − 14.6·9-s + (−8.08 + 59.4i)10-s − 41.3·11-s − 134. i·12-s + 13i·13-s + 64.6·14-s + (9.72 − 71.5i)15-s + 202.·16-s − 39.1i·17-s + ⋯
L(s)  = 1  − 1.89i·2-s + 1.24i·3-s − 2.59·4-s + (−0.990 − 0.134i)5-s + 2.35·6-s + 0.650i·7-s + 3.03i·8-s − 0.543·9-s + (−0.255 + 1.88i)10-s − 1.13·11-s − 3.23i·12-s + 0.277i·13-s + 1.23·14-s + (0.167 − 1.23i)15-s + 3.15·16-s − 0.559i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.134 - 0.990i$
Analytic conductor: \(3.83512\)
Root analytic conductor: \(1.95834\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :3/2),\ 0.134 - 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.193019 + 0.168559i\)
\(L(\frac12)\) \(\approx\) \(0.193019 + 0.168559i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (11.0 + 1.50i)T \)
13 \( 1 - 13iT \)
good2 \( 1 + 5.36iT - 8T^{2} \)
3 \( 1 - 6.45iT - 27T^{2} \)
7 \( 1 - 12.0iT - 343T^{2} \)
11 \( 1 + 41.3T + 1.33e3T^{2} \)
17 \( 1 + 39.1iT - 4.91e3T^{2} \)
19 \( 1 + 144.T + 6.85e3T^{2} \)
23 \( 1 + 97.2iT - 1.21e4T^{2} \)
29 \( 1 - 78.8T + 2.43e4T^{2} \)
31 \( 1 + 213.T + 2.97e4T^{2} \)
37 \( 1 - 325. iT - 5.06e4T^{2} \)
41 \( 1 - 304.T + 6.89e4T^{2} \)
43 \( 1 + 5.82iT - 7.95e4T^{2} \)
47 \( 1 - 67.9iT - 1.03e5T^{2} \)
53 \( 1 + 108. iT - 1.48e5T^{2} \)
59 \( 1 + 109.T + 2.05e5T^{2} \)
61 \( 1 + 342.T + 2.26e5T^{2} \)
67 \( 1 + 453. iT - 3.00e5T^{2} \)
71 \( 1 + 323.T + 3.57e5T^{2} \)
73 \( 1 - 61.8iT - 3.89e5T^{2} \)
79 \( 1 + 581.T + 4.93e5T^{2} \)
83 \( 1 - 1.49e3iT - 5.71e5T^{2} \)
89 \( 1 + 447.T + 7.04e5T^{2} \)
97 \( 1 - 784. iT - 9.12e5T^{2} \)
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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

−14.61786931347782978836351410220, −13.00598713168252394555985184278, −12.19762824917659801710406074279, −11.02223255252456637796053911846, −10.47609320118474066270852709391, −9.241932907618606675708029869940, −8.344793033719653656139232843552, −4.96961929719098625169367455169, −4.12135750820170602658848090816, −2.71763700082823939995611106836, 0.17305121197434823559371484425, 4.21784649381377708517044663556, 5.91988271718113719959853071155, 7.20504056955224390911766846828, 7.66281472211319377720826317217, 8.617527103651948702676413033137, 10.61131689714456532507267112733, 12.66275026251041776774891192410, 13.14035551764852208610636226411, 14.34459638718035709825319382640

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