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Properties

Label 2-45-5.2-c4-0-0
Degree $2$
Conductor $45$
Sign $0.958 - 0.283i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.80 − 3.80i)2-s + 12.9i·4-s + (−6.56 + 24.1i)5-s + (16.6 + 16.6i)7-s + (−11.6 + 11.6i)8-s + (116. − 66.7i)10-s + 215.·11-s + (−29.9 + 29.9i)13-s − 126. i·14-s + 295.·16-s + (3.97 + 3.97i)17-s + 604. i·19-s + (−311. − 84.9i)20-s + (−818. − 818. i)22-s + (−376. + 376. i)23-s + ⋯
L(s)  = 1  + (−0.950 − 0.950i)2-s + 0.808i·4-s + (−0.262 + 0.964i)5-s + (0.339 + 0.339i)7-s + (−0.182 + 0.182i)8-s + (1.16 − 0.667i)10-s + 1.77·11-s + (−0.177 + 0.177i)13-s − 0.644i·14-s + 1.15·16-s + (0.0137 + 0.0137i)17-s + 1.67i·19-s + (−0.779 − 0.212i)20-s + (−1.69 − 1.69i)22-s + (−0.711 + 0.711i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.958 - 0.283i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.958 - 0.283i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.831734 + 0.120471i\)
\(L(\frac12)\) \(\approx\) \(0.831734 + 0.120471i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (6.56 - 24.1i)T \)
good2 \( 1 + (3.80 + 3.80i)T + 16iT^{2} \)
7 \( 1 + (-16.6 - 16.6i)T + 2.40e3iT^{2} \)
11 \( 1 - 215.T + 1.46e4T^{2} \)
13 \( 1 + (29.9 - 29.9i)T - 2.85e4iT^{2} \)
17 \( 1 + (-3.97 - 3.97i)T + 8.35e4iT^{2} \)
19 \( 1 - 604. iT - 1.30e5T^{2} \)
23 \( 1 + (376. - 376. i)T - 2.79e5iT^{2} \)
29 \( 1 - 624. iT - 7.07e5T^{2} \)
31 \( 1 - 263.T + 9.23e5T^{2} \)
37 \( 1 + (-979. - 979. i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.57e3T + 2.82e6T^{2} \)
43 \( 1 + (30.8 - 30.8i)T - 3.41e6iT^{2} \)
47 \( 1 + (1.78e3 + 1.78e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (707. - 707. i)T - 7.89e6iT^{2} \)
59 \( 1 + 1.36e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.98e3T + 1.38e7T^{2} \)
67 \( 1 + (986. + 986. i)T + 2.01e7iT^{2} \)
71 \( 1 + 68.0T + 2.54e7T^{2} \)
73 \( 1 + (-2.47e3 + 2.47e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 6.04e3iT - 3.89e7T^{2} \)
83 \( 1 + (-5.15e3 + 5.15e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 7.02e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.08e4 - 1.08e4i)T + 8.85e7iT^{2} \)
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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

−14.87664250505277829600095515292, −14.24794480217943376055480510384, −12.05705809649070947308145302902, −11.55023607343392023280095318713, −10.30485465520781565862380753565, −9.288549895813705477385506835756, −7.937400100767493557076628656245, −6.23241761615639168187215979850, −3.58548807063401939161266522531, −1.71143705428645798906169085396, 0.798289300451765372052626030605, 4.34401598142625114555860784365, 6.29602641274894401920615070815, 7.60696475657327907921158999599, 8.788220463060621843873178006167, 9.556828985538710610858438676394, 11.44141749130610746891047575294, 12.64636843105268172224473904748, 14.19612791544485433007779403774, 15.38017118794153135043631292736

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