Properties

Label 2-65-65.2-c1-0-3
Degree $2$
Conductor $65$
Sign $0.320 + 0.947i$
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  + (−0.137 − 0.237i)2-s + (−2.28 − 0.611i)3-s + (0.962 − 1.66i)4-s + (1.69 − 1.45i)5-s + (0.168 + 0.627i)6-s + (−0.334 − 0.193i)7-s − 1.07·8-s + (2.23 + 1.29i)9-s + (−0.579 − 0.204i)10-s + (−1.12 + 4.21i)11-s + (−3.21 + 3.21i)12-s + (3.34 + 1.35i)13-s + 0.106i·14-s + (−4.76 + 2.27i)15-s + (−1.77 − 3.07i)16-s + (0.510 + 1.90i)17-s + ⋯
L(s)  = 1  + (−0.0971 − 0.168i)2-s + (−1.31 − 0.353i)3-s + (0.481 − 0.833i)4-s + (0.759 − 0.650i)5-s + (0.0686 + 0.256i)6-s + (−0.126 − 0.0729i)7-s − 0.381·8-s + (0.745 + 0.430i)9-s + (−0.183 − 0.0646i)10-s + (−0.340 + 1.27i)11-s + (−0.928 + 0.928i)12-s + (0.926 + 0.376i)13-s + 0.0283i·14-s + (−1.23 + 0.588i)15-s + (−0.444 − 0.769i)16-s + (0.123 + 0.462i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.320 + 0.947i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.320 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577118 - 0.414081i\)
\(L(\frac12)\) \(\approx\) \(0.577118 - 0.414081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.69 + 1.45i)T \)
13 \( 1 + (-3.34 - 1.35i)T \)
good2 \( 1 + (0.137 + 0.237i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (2.28 + 0.611i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.334 + 0.193i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.12 - 4.21i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.510 - 1.90i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-4.83 + 1.29i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.0863 + 0.322i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (7.07 - 4.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.54 - 2.54i)T - 31iT^{2} \)
37 \( 1 + (-4.17 + 2.41i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.49 - 1.20i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.58 - 1.76i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 9.83iT - 47T^{2} \)
53 \( 1 + (7.17 - 7.17i)T - 53iT^{2} \)
59 \( 1 + (-0.628 - 2.34i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.32 - 9.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.18 - 5.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.12 + 4.20i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 6.08T + 73T^{2} \)
79 \( 1 + 3.34iT - 79T^{2} \)
83 \( 1 + 5.18iT - 83T^{2} \)
89 \( 1 + (4.82 + 1.29i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.37 - 12.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.74592593090985261272942726430, −13.32066426088197637839160566944, −12.33674686867864104054600628226, −11.30292886313383061997862931559, −10.30681222609611674974355872967, −9.267728270028469334772485838752, −7.06070138917915795895077903995, −5.94766677779297558110428746518, −5.06719445927738014698327155025, −1.53387025839927706538176263620, 3.26631185424053604873635933456, 5.62272849505041732032760111651, 6.33695848563938735633070110731, 7.87251593343471570771739336366, 9.584828213109349033324054074019, 11.13544594804224906876633999499, 11.26053359682759641450474352674, 12.82478449603887659210313967975, 13.89926329216979603686063447580, 15.58418156440501570184463056098

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