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Properties

Label 2-45-45.13-c2-0-1
Degree $2$
Conductor $45$
Sign $0.678 - 0.734i$
Analytic cond. $1.22616$
Root an. cond. $1.10732$
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  + (−2.38 − 0.639i)2-s + (2.13 + 2.11i)3-s + (1.82 + 1.05i)4-s + (−1.83 + 4.64i)5-s + (−3.73 − 6.40i)6-s + (10.3 + 2.77i)7-s + (3.30 + 3.30i)8-s + (0.0915 + 8.99i)9-s + (7.36 − 9.92i)10-s + (−5.66 − 9.81i)11-s + (1.66 + 6.09i)12-s + (−9.44 + 2.53i)13-s + (−22.9 − 13.2i)14-s + (−13.7 + 6.02i)15-s + (−9.99 − 17.3i)16-s + (8.05 − 8.05i)17-s + ⋯
L(s)  = 1  + (−1.19 − 0.319i)2-s + (0.710 + 0.703i)3-s + (0.456 + 0.263i)4-s + (−0.367 + 0.929i)5-s + (−0.623 − 1.06i)6-s + (1.48 + 0.396i)7-s + (0.413 + 0.413i)8-s + (0.0101 + 0.999i)9-s + (0.736 − 0.992i)10-s + (−0.514 − 0.891i)11-s + (0.138 + 0.508i)12-s + (−0.726 + 0.194i)13-s + (−1.63 − 0.946i)14-s + (−0.915 + 0.401i)15-s + (−0.624 − 1.08i)16-s + (0.473 − 0.473i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ 0.678 - 0.734i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.702321 + 0.307380i\)
\(L(\frac12)\) \(\approx\) \(0.702321 + 0.307380i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.13 - 2.11i)T \)
5 \( 1 + (1.83 - 4.64i)T \)
good2 \( 1 + (2.38 + 0.639i)T + (3.46 + 2i)T^{2} \)
7 \( 1 + (-10.3 - 2.77i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (5.66 + 9.81i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (9.44 - 2.53i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (-8.05 + 8.05i)T - 289iT^{2} \)
19 \( 1 + 3.73iT - 361T^{2} \)
23 \( 1 + (-13.4 + 3.60i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (-28.7 + 16.6i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-7.67 + 13.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-13.7 + 13.7i)T - 1.36e3iT^{2} \)
41 \( 1 + (-4.87 + 8.44i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (10.2 - 38.3i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-1.68 - 0.451i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (45.3 + 45.3i)T + 2.80e3iT^{2} \)
59 \( 1 + (10.2 + 5.91i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (28.0 + 48.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (12.7 + 47.4i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 22.2T + 5.04e3T^{2} \)
73 \( 1 + (-37.2 - 37.2i)T + 5.32e3iT^{2} \)
79 \( 1 + (105. - 60.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (28.1 - 105. i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 16.0iT - 7.92e3T^{2} \)
97 \( 1 + (-172. - 46.1i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−15.76357053986476106890972074450, −14.61504154045804360044217315386, −13.97145916894152035188729138747, −11.48935360194219106967914282128, −10.83517715758167479543798197964, −9.699069458849175636109902018824, −8.367241163460656217874045605788, −7.70264135193135657190325197690, −4.89221226273715963850887067849, −2.57528348622000585511177947963, 1.39364974556289691310302733006, 4.65017054995212818616966040275, 7.37432027292541943673298491302, 7.979175674995482505874523391594, 8.901008898537101179982059200976, 10.27162355723260003636831515219, 12.02637364466375020517964539478, 13.08701334955531399264949983834, 14.46527991682760582600980884850, 15.54761585610479193226880961340

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