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Properties

Label 2-65-13.11-c2-0-6
Degree $2$
Conductor $65$
Sign $0.495 + 0.868i$
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  + (−0.947 + 3.53i)2-s + (−2.34 − 4.05i)3-s + (−8.14 − 4.70i)4-s + (1.58 + 1.58i)5-s + (16.5 − 4.43i)6-s + (−2.82 − 10.5i)7-s + (13.9 − 13.9i)8-s + (−6.46 + 11.1i)9-s + (−7.09 + 4.09i)10-s + (−2.43 − 0.653i)11-s + 44.0i·12-s + (−12.3 − 3.90i)13-s + 39.9·14-s + (2.70 − 10.1i)15-s + (17.4 + 30.1i)16-s + (−12.7 − 7.38i)17-s + ⋯
L(s)  = 1  + (−0.473 + 1.76i)2-s + (−0.780 − 1.35i)3-s + (−2.03 − 1.17i)4-s + (0.316 + 0.316i)5-s + (2.75 − 0.739i)6-s + (−0.403 − 1.50i)7-s + (1.74 − 1.74i)8-s + (−0.718 + 1.24i)9-s + (−0.709 + 0.409i)10-s + (−0.221 − 0.0594i)11-s + 3.67i·12-s + (−0.953 − 0.300i)13-s + 2.85·14-s + (0.180 − 0.674i)15-s + (1.08 + 1.88i)16-s + (−0.752 − 0.434i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.495 + 0.868i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ 0.495 + 0.868i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.367803 - 0.213540i\)
\(L(\frac12)\) \(\approx\) \(0.367803 - 0.213540i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.58 - 1.58i)T \)
13 \( 1 + (12.3 + 3.90i)T \)
good2 \( 1 + (0.947 - 3.53i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (2.34 + 4.05i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (2.82 + 10.5i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (2.43 + 0.653i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (12.7 + 7.38i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-9.12 + 2.44i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-9.30 + 5.37i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-11.5 - 20.0i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (2.04 + 2.04i)T + 961iT^{2} \)
37 \( 1 + (-14.2 - 3.82i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-9.33 + 34.8i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-22.9 - 13.2i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (19.8 - 19.8i)T - 2.20e3iT^{2} \)
53 \( 1 - 22.5T + 2.80e3T^{2} \)
59 \( 1 + (29.1 + 108. i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-41.4 + 71.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-27.5 + 102. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-26.4 + 7.09i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-21.9 + 21.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 148.T + 6.24e3T^{2} \)
83 \( 1 + (-24.8 - 24.8i)T + 6.88e3iT^{2} \)
89 \( 1 + (-124. - 33.3i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (165. - 44.3i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−14.35735953087374253095485439662, −13.62282388195411665173535872091, −12.75587188758638339968465532068, −10.87153186510241106693792769590, −9.602611991700748633753240040700, −7.81352177196249899028936376541, −7.07805547955016051263168520386, −6.46613301669889262423924274136, −5.03784285545488818279953916445, −0.49211103480054965451252702868, 2.60931758938503876067342674686, 4.37411411716268350004847511896, 5.56261367233737528615781017084, 8.756130207838172939636888098091, 9.525871482736283725968561526852, 10.22825032249307996902200258071, 11.44959813425640413304363375468, 12.08527748407913640861643542438, 13.13070048504248501292249439813, 14.94999275315960100935666599350

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