Properties

Label 2-65-65.9-c1-0-4
Degree $2$
Conductor $65$
Sign $0.796 - 0.604i$
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  + (2.20 + 1.27i)2-s + (−1.86 − 1.07i)3-s + (2.24 + 3.88i)4-s + (−0.817 − 2.08i)5-s + (−2.74 − 4.74i)6-s + (−2.54 + 1.46i)7-s + 6.31i·8-s + (0.817 + 1.41i)9-s + (0.846 − 5.62i)10-s + (0.317 − 0.550i)11-s − 9.64i·12-s + (3.60 − 0.0716i)13-s − 7.48·14-s + (−0.716 + 4.76i)15-s + (−3.55 + 6.16i)16-s + (−1.05 + 0.611i)17-s + ⋯
L(s)  = 1  + (1.55 + 0.900i)2-s + (−1.07 − 0.621i)3-s + (1.12 + 1.94i)4-s + (−0.365 − 0.930i)5-s + (−1.11 − 1.93i)6-s + (−0.961 + 0.555i)7-s + 2.23i·8-s + (0.272 + 0.472i)9-s + (0.267 − 1.78i)10-s + (0.0957 − 0.165i)11-s − 2.78i·12-s + (0.999 − 0.0198i)13-s − 1.99·14-s + (−0.184 + 1.22i)15-s + (−0.889 + 1.54i)16-s + (−0.257 + 0.148i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.796 - 0.604i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.796 - 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23911 + 0.417214i\)
\(L(\frac12)\) \(\approx\) \(1.23911 + 0.417214i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.817 + 2.08i)T \)
13 \( 1 + (-3.60 + 0.0716i)T \)
good2 \( 1 + (-2.20 - 1.27i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.86 + 1.07i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.54 - 1.46i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.317 + 0.550i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.05 - 0.611i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.682 - 1.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.86 - 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.96T + 31T^{2} \)
37 \( 1 + (1.05 + 0.611i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.98 + 8.62i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.18 + 0.683i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.16iT - 47T^{2} \)
53 \( 1 + 0.642iT - 53T^{2} \)
59 \( 1 + (-3.79 - 6.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.95 + 4.01i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.31 + 2.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + 1.03T + 79T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + (6.27 - 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.8 - 7.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−15.17105496518101060015345589398, −13.60404708760317876594399235840, −12.79454843159501921500853351561, −12.25519325420685023646553764340, −11.28089107109370560112882827898, −8.836623480869749635479402107278, −7.24978069514871038636930041891, −6.08111831536227300301347688791, −5.43924391818933259494223512527, −3.76997695851957584202444893920, 3.23694847152146753588120885048, 4.36765884739908055310249080766, 5.86142569462599755827231062822, 6.80204784739838765136144414895, 9.881505728533565667560510139410, 10.93195764842757882743898550499, 11.22609257040916409673136678930, 12.53000259108757617207610306448, 13.52602767205090511422753487331, 14.60638073227964489770011197838

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