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Properties

Label 2-45-45.29-c4-0-21
Degree $2$
Conductor $45$
Sign $-0.978 + 0.208i$
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  + (3.75 − 6.50i)2-s + (−3.49 − 8.29i)3-s + (−20.1 − 34.9i)4-s + (22.9 + 10.0i)5-s + (−67.0 − 8.43i)6-s + (60.4 + 34.9i)7-s − 182.·8-s + (−56.6 + 57.9i)9-s + (151. − 111. i)10-s + (−45.1 − 26.0i)11-s + (−219. + 289. i)12-s + (104. − 60.2i)13-s + (454. − 262. i)14-s + (3.08 − 224. i)15-s + (−363. + 629. i)16-s + 80.3·17-s + ⋯
L(s)  = 1  + (0.938 − 1.62i)2-s + (−0.387 − 0.921i)3-s + (−1.26 − 2.18i)4-s + (0.916 + 0.400i)5-s + (−1.86 − 0.234i)6-s + (1.23 + 0.712i)7-s − 2.85·8-s + (−0.699 + 0.715i)9-s + (1.51 − 1.11i)10-s + (−0.373 − 0.215i)11-s + (−1.52 + 2.00i)12-s + (0.617 − 0.356i)13-s + (2.31 − 1.33i)14-s + (0.0136 − 0.999i)15-s + (−1.42 + 2.45i)16-s + 0.278·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.978 + 0.208i$
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: \((2,\ 45,\ (\ :2),\ -0.978 + 0.208i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.240770 - 2.28324i\)
\(L(\frac12)\) \(\approx\) \(0.240770 - 2.28324i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.49 + 8.29i)T \)
5 \( 1 + (-22.9 - 10.0i)T \)
good2 \( 1 + (-3.75 + 6.50i)T + (-8 - 13.8i)T^{2} \)
7 \( 1 + (-60.4 - 34.9i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (45.1 + 26.0i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-104. + 60.2i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 80.3T + 8.35e4T^{2} \)
19 \( 1 + 254.T + 1.30e5T^{2} \)
23 \( 1 + (125. + 216. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-250. - 144. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (11.0 + 19.1i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 2.47e3iT - 1.87e6T^{2} \)
41 \( 1 + (-421. + 243. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-86.4 - 49.9i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.73e3 + 3.01e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 2.91e3T + 7.89e6T^{2} \)
59 \( 1 + (1.51e3 - 875. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.79e3 - 4.83e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.32e3 + 1.92e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 988. iT - 2.54e7T^{2} \)
73 \( 1 - 6.01e3iT - 2.83e7T^{2} \)
79 \( 1 + (31.6 - 54.8i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-727. + 1.26e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 1.28e4iT - 6.27e7T^{2} \)
97 \( 1 + (-6.34e3 - 3.66e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−14.01582405615724782744883072306, −13.21978166989832576254231454778, −12.17077627619765917689951569885, −11.20059772096881380748300741851, −10.37631384880634569220671718340, −8.550732541281100934015721179453, −6.05737308066031652974605561215, −5.07283218798692062881593707561, −2.61682647533642484972924197846, −1.46516352558325412539928246808, 4.17780080614338658526595199104, 5.12363098142331681870153654774, 6.22198588332098193814356401954, 7.908104182921690523464369394590, 9.168118464528785006914333212985, 10.87931229144250073708701332372, 12.55934112505639339389079762316, 13.91047300660016031820664261144, 14.43513664348043460110499658374, 15.62108589379416650812113599531

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