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Properties

Label 2-45-45.29-c4-0-19
Degree $2$
Conductor $45$
Sign $-0.647 + 0.762i$
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  + (3.12 − 5.41i)2-s + (7.73 − 4.59i)3-s + (−11.5 − 19.9i)4-s + (−24.9 − 0.765i)5-s + (−0.705 − 56.2i)6-s + (10.1 + 5.84i)7-s − 43.8·8-s + (38.7 − 71.1i)9-s + (−82.1 + 132. i)10-s + (141. + 81.6i)11-s + (−180. − 101. i)12-s + (−48.3 + 27.9i)13-s + (63.2 − 36.5i)14-s + (−196. + 108. i)15-s + (47.1 − 81.6i)16-s + 256.·17-s + ⋯
L(s)  = 1  + (0.780 − 1.35i)2-s + (0.859 − 0.510i)3-s + (−0.719 − 1.24i)4-s + (−0.999 − 0.0306i)5-s + (−0.0195 − 1.56i)6-s + (0.206 + 0.119i)7-s − 0.685·8-s + (0.478 − 0.878i)9-s + (−0.821 + 1.32i)10-s + (1.16 + 0.675i)11-s + (−1.25 − 0.703i)12-s + (−0.286 + 0.165i)13-s + (0.322 − 0.186i)14-s + (−0.874 + 0.484i)15-s + (0.184 − 0.318i)16-s + 0.888·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.647 + 0.762i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ -0.647 + 0.762i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.05868 - 2.28680i\)
\(L(\frac12)\) \(\approx\) \(1.05868 - 2.28680i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.73 + 4.59i)T \)
5 \( 1 + (24.9 + 0.765i)T \)
good2 \( 1 + (-3.12 + 5.41i)T + (-8 - 13.8i)T^{2} \)
7 \( 1 + (-10.1 - 5.84i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-141. - 81.6i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (48.3 - 27.9i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 256.T + 8.35e4T^{2} \)
19 \( 1 + 618.T + 1.30e5T^{2} \)
23 \( 1 + (-329. - 570. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-122. - 70.8i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-300. - 519. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 751. iT - 1.87e6T^{2} \)
41 \( 1 + (2.03e3 - 1.17e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.71e3 + 993. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (433. - 751. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 3.19e3T + 7.89e6T^{2} \)
59 \( 1 + (2.39e3 - 1.38e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-48.4 + 83.8i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.75e3 - 2.16e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 8.11e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.09e3iT - 2.83e7T^{2} \)
79 \( 1 + (929. - 1.61e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-6.08e3 + 1.05e4i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 1.06e4iT - 6.27e7T^{2} \)
97 \( 1 + (5.62e3 + 3.24e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−14.53319995032433050815968006315, −13.21940283637006230702442238825, −12.22522369037193078030273708007, −11.61871123764755786933041723861, −10.01951500905297616731623997087, −8.574791538657483862780757615272, −7.05708850971936293074893839234, −4.45136790629671336548771311850, −3.31165033397019302445996458194, −1.55296504572577284001884551691, 3.64302347550044741662093417592, 4.68411261704705999439975810454, 6.60208218543697015331804915811, 7.920543903265408891303977110807, 8.745611000762155517914491576805, 10.71158078525035151602461413463, 12.39170709379632275732665984726, 13.71728470320340974919804114441, 14.82284500529230941141247316580, 15.02833988156695523576258803926

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