Properties

Label 2-65-65.4-c1-0-2
Degree $2$
Conductor $65$
Sign $0.998 - 0.0496i$
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  + (0.228 − 0.395i)2-s + (0.866 + 0.5i)3-s + (0.895 + 1.55i)4-s + (−2.18 − 0.456i)5-s + (0.395 − 0.228i)6-s + (−0.866 − 1.5i)7-s + 1.73·8-s + (−1 − 1.73i)9-s + (−0.680 + 0.761i)10-s + (−2.29 − 1.32i)11-s + 1.79i·12-s + (3.46 + i)13-s − 0.791·14-s + (−1.66 − 1.49i)15-s + (−1.39 + 2.41i)16-s + (−3.96 + 2.29i)17-s + ⋯
L(s)  = 1  + (0.161 − 0.279i)2-s + (0.499 + 0.288i)3-s + (0.447 + 0.775i)4-s + (−0.978 − 0.204i)5-s + (0.161 − 0.0932i)6-s + (−0.327 − 0.566i)7-s + 0.612·8-s + (−0.333 − 0.577i)9-s + (−0.215 + 0.240i)10-s + (−0.690 − 0.398i)11-s + 0.517i·12-s + (0.960 + 0.277i)13-s − 0.211·14-s + (−0.430 − 0.384i)15-s + (−0.348 + 0.604i)16-s + (−0.962 + 0.555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.998 - 0.0496i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.998 - 0.0496i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02275 + 0.0253843i\)
\(L(\frac12)\) \(\approx\) \(1.02275 + 0.0253843i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.18 + 0.456i)T \)
13 \( 1 + (-3.46 - i)T \)
good2 \( 1 + (-0.228 + 0.395i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.866 - 0.5i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.866 + 1.5i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.29 + 1.32i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.96 - 2.29i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.96 - 2.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.29 + 3.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.20iT - 31T^{2} \)
37 \( 1 + (-3.96 + 6.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.29 - 1.32i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.16 - 5.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.82T + 47T^{2} \)
53 \( 1 + 7.58iT - 53T^{2} \)
59 \( 1 + (12.0 - 6.97i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.708 - 1.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.504 - 0.873i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.08 + 3.51i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 6.01T + 83T^{2} \)
89 \( 1 + (-8.29 - 4.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.70 + 9.87i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.12786442662525226043117884281, −13.59610760463297749621082631833, −12.73785212646342377897351992444, −11.51723502988010633281204320021, −10.70728630722250951887977646561, −8.849400630153204233572009066947, −8.010867862262315465823982381317, −6.63245746008387717216145898156, −4.16201228896303372136314573058, −3.20822318418770939066641300386, 2.67516543151767637733438152719, 4.90972446654755545152386280136, 6.50253374052466875181641935995, 7.70782616002010372482731212264, 8.874224455871232030128177799755, 10.60413139986680490071109025125, 11.37597480694708182434760309246, 12.89074929844707762337727772137, 13.89773740875203017955071674250, 15.21646402826567298278817069145

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