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Properties

Label 2-65-5.4-c1-0-2
Degree $2$
Conductor $65$
Sign $0.970 + 0.241i$
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  − 1.53i·2-s + 3.17i·3-s − 0.369·4-s + (0.539 − 2.17i)5-s + 4.87·6-s + 1.70i·7-s − 2.51i·8-s − 7.04·9-s + (−3.34 − 0.829i)10-s − 2.53·11-s − 1.17i·12-s − i·13-s + 2.63·14-s + (6.87 + 1.70i)15-s − 4.60·16-s − 0.921i·17-s + ⋯
L(s)  = 1  − 1.08i·2-s + 1.83i·3-s − 0.184·4-s + (0.241 − 0.970i)5-s + 1.99·6-s + 0.646i·7-s − 0.887i·8-s − 2.34·9-s + (−1.05 − 0.262i)10-s − 0.765·11-s − 0.337i·12-s − 0.277i·13-s + 0.703·14-s + (1.77 + 0.441i)15-s − 1.15·16-s − 0.223i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.970 + 0.241i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.929835 - 0.113785i\)
\(L(\frac12)\) \(\approx\) \(0.929835 - 0.113785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.539 + 2.17i)T \)
13 \( 1 + iT \)
good2 \( 1 + 1.53iT - 2T^{2} \)
3 \( 1 - 3.17iT - 3T^{2} \)
7 \( 1 - 1.70iT - 7T^{2} \)
11 \( 1 + 2.53T + 11T^{2} \)
17 \( 1 + 0.921iT - 17T^{2} \)
19 \( 1 - 0.539T + 19T^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 - 5.12T + 29T^{2} \)
31 \( 1 - 0.879T + 31T^{2} \)
37 \( 1 - 6.04iT - 37T^{2} \)
41 \( 1 - 1.26T + 41T^{2} \)
43 \( 1 - 6.43iT - 43T^{2} \)
47 \( 1 + 5.70iT - 47T^{2} \)
53 \( 1 - 8.49iT - 53T^{2} \)
59 \( 1 - 4.72T + 59T^{2} \)
61 \( 1 - 8.04T + 61T^{2} \)
67 \( 1 + 7.86iT - 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + 1.95iT - 73T^{2} \)
79 \( 1 + 0.496T + 79T^{2} \)
83 \( 1 - 8.63iT - 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 5.91iT - 97T^{2} \)
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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

−15.24787171750382351667803247251, −13.60263918922203102099225860925, −12.29635248254585923843770996969, −11.34580426482071455207166203577, −10.23293480800230196875249855276, −9.563432298274384761159861730566, −8.496681785240381725133340313688, −5.59756719719464072192820416782, −4.46064862408598352609754857070, −2.90935368787153890143350408498, 2.43709656383644875406456838039, 5.77023416205514375906965964488, 6.79585552462992882466771779622, 7.38224040327350772358246039198, 8.379246291915024006143123350513, 10.62855357115481254865513646324, 11.74650622219940963261757845826, 13.10735196488875680148765534523, 14.00644552960605766915533717822, 14.62888949250124449935920431761

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