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Properties

Label 2-65-65.38-c2-0-11
Degree $2$
Conductor $65$
Sign $0.275 - 0.961i$
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  + (−2.66 − 2.66i)2-s + (−2.12 − 2.12i)3-s + 10.1i·4-s + (−4.54 − 2.09i)5-s + 11.3i·6-s + (2.95 + 2.95i)7-s + (16.4 − 16.4i)8-s + 0.0323i·9-s + (6.51 + 17.6i)10-s + 8.43i·11-s + (21.6 − 21.6i)12-s + (−9.88 + 8.44i)13-s − 15.7i·14-s + (5.19 + 14.0i)15-s − 47.1·16-s + (−9.34 + 9.34i)17-s + ⋯
L(s)  = 1  + (−1.33 − 1.33i)2-s + (−0.708 − 0.708i)3-s + 2.54i·4-s + (−0.908 − 0.418i)5-s + 1.88i·6-s + (0.421 + 0.421i)7-s + (2.06 − 2.06i)8-s + 0.00359i·9-s + (0.651 + 1.76i)10-s + 0.767i·11-s + (1.80 − 1.80i)12-s + (−0.760 + 0.649i)13-s − 1.12i·14-s + (0.346 + 0.939i)15-s − 2.94·16-s + (−0.549 + 0.549i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ 0.275 - 0.961i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00134578 + 0.00101429i\)
\(L(\frac12)\) \(\approx\) \(0.00134578 + 0.00101429i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.54 + 2.09i)T \)
13 \( 1 + (9.88 - 8.44i)T \)
good2 \( 1 + (2.66 + 2.66i)T + 4iT^{2} \)
3 \( 1 + (2.12 + 2.12i)T + 9iT^{2} \)
7 \( 1 + (-2.95 - 2.95i)T + 49iT^{2} \)
11 \( 1 - 8.43iT - 121T^{2} \)
17 \( 1 + (9.34 - 9.34i)T - 289iT^{2} \)
19 \( 1 + 12.7T + 361T^{2} \)
23 \( 1 + (28.0 + 28.0i)T + 529iT^{2} \)
29 \( 1 - 32.7iT - 841T^{2} \)
31 \( 1 + 36.2iT - 961T^{2} \)
37 \( 1 + (-17.1 - 17.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 26.7iT - 1.68e3T^{2} \)
43 \( 1 + (34.4 + 34.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (9.09 + 9.09i)T + 2.20e3iT^{2} \)
53 \( 1 + (-31.0 - 31.0i)T + 2.80e3iT^{2} \)
59 \( 1 + 68.4T + 3.48e3T^{2} \)
61 \( 1 + 30.3T + 3.72e3T^{2} \)
67 \( 1 + (5.29 + 5.29i)T + 4.48e3iT^{2} \)
71 \( 1 + 2.71iT - 5.04e3T^{2} \)
73 \( 1 + (29.3 - 29.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 12.1iT - 6.24e3T^{2} \)
83 \( 1 + (3.27 - 3.27i)T - 6.88e3iT^{2} \)
89 \( 1 + 68.3T + 7.92e3T^{2} \)
97 \( 1 + (-51.9 - 51.9i)T + 9.40e3iT^{2} \)
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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

−12.84593382398430740638230558834, −12.16375362393841906088394036946, −11.67619246132775874947098653709, −10.52669683021708532236779298968, −9.125438499742066640183789922326, −8.114257015208009374088432381950, −6.94740817316902363701417818197, −4.26691882527784982802209492212, −1.94798964310347747053721463676, −0.00248442147202340548504194465, 4.69179238155416898585358388298, 6.04560361595719222066099286212, 7.47617711254723579704341165162, 8.236230112075606169587075681487, 9.793990184345056650434235593515, 10.71230555296395248135331407702, 11.49172666221858221643185879379, 13.91914097699901284935100239054, 15.04689026251565901054198601490, 15.79689079941501870290522811095

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