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Properties

Label 2-65-65.33-c1-0-3
Degree $2$
Conductor $65$
Sign $0.525 - 0.850i$
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  + (−0.915 + 1.58i)2-s + (1.91 − 0.512i)3-s + (−0.677 − 1.17i)4-s + (1.45 − 1.69i)5-s + (−0.939 + 3.50i)6-s + (−3.06 + 1.76i)7-s − 1.18·8-s + (0.803 − 0.463i)9-s + (1.36 + 3.86i)10-s + (−1.00 − 3.74i)11-s + (−1.89 − 1.89i)12-s + (0.573 + 3.55i)13-s − 6.48i·14-s + (1.91 − 3.99i)15-s + (2.43 − 4.22i)16-s + (0.524 − 1.95i)17-s + ⋯
L(s)  = 1  + (−0.647 + 1.12i)2-s + (1.10 − 0.296i)3-s + (−0.338 − 0.586i)4-s + (0.650 − 0.759i)5-s + (−0.383 + 1.43i)6-s + (−1.15 + 0.668i)7-s − 0.417·8-s + (0.267 − 0.154i)9-s + (0.430 + 1.22i)10-s + (−0.302 − 1.12i)11-s + (−0.548 − 0.548i)12-s + (0.158 + 0.987i)13-s − 1.73i·14-s + (0.494 − 1.03i)15-s + (0.609 − 1.05i)16-s + (0.127 − 0.474i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755460 + 0.421510i\)
\(L(\frac12)\) \(\approx\) \(0.755460 + 0.421510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.45 + 1.69i)T \)
13 \( 1 + (-0.573 - 3.55i)T \)
good2 \( 1 + (0.915 - 1.58i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.91 + 0.512i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (3.06 - 1.76i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.00 + 3.74i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.524 + 1.95i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.518 + 0.139i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.0788 - 0.294i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.71 - 0.988i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.13 - 4.13i)T + 31iT^{2} \)
37 \( 1 + (-4.69 - 2.70i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.649 + 0.174i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (8.51 + 2.28i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 9.75iT - 47T^{2} \)
53 \( 1 + (-3.16 - 3.16i)T + 53iT^{2} \)
59 \( 1 + (3.14 - 11.7i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.44 - 2.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.14 - 1.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.19 - 4.46i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 1.59iT - 79T^{2} \)
83 \( 1 + 7.57iT - 83T^{2} \)
89 \( 1 + (4.54 - 1.21i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (8.91 + 15.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.29444299726111024841273776883, −14.00190902615929797659815827529, −13.32855559005402164075033193527, −12.01506509370145817684837444914, −9.744731293682046034845722419143, −8.864627325720615846041824234345, −8.385545879766587682997595913045, −6.77571061513121592632839743525, −5.67675069296987767716311391101, −2.91048034431394291908040848436, 2.49764371878893874530695971166, 3.52072664395654940050195391528, 6.35065393113102818095214357936, 8.000765378017471032875171220300, 9.645039100334906023632134896233, 9.854852711903971344877447822632, 10.87427119346016831208139223179, 12.59622291525836117976227181742, 13.45403956872348334933972661273, 14.74682311237727137108239334856

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