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Properties

Label 2-45-45.29-c4-0-7
Degree $2$
Conductor $45$
Sign $0.898 - 0.439i$
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  + (2.30 − 3.99i)2-s + (−4.25 + 7.93i)3-s + (−2.66 − 4.61i)4-s + (1.07 + 24.9i)5-s + (21.9 + 35.3i)6-s + (45.6 + 26.3i)7-s + 49.2·8-s + (−44.8 − 67.4i)9-s + (102. + 53.3i)10-s + (−1.90 − 1.09i)11-s + (47.9 − 1.51i)12-s + (−226. + 131. i)13-s + (210. − 121. i)14-s + (−202. − 97.6i)15-s + (156. − 270. i)16-s + 487.·17-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)2-s + (−0.472 + 0.881i)3-s + (−0.166 − 0.288i)4-s + (0.0428 + 0.999i)5-s + (0.608 + 0.981i)6-s + (0.930 + 0.537i)7-s + 0.769·8-s + (−0.553 − 0.832i)9-s + (1.02 + 0.533i)10-s + (−0.0157 − 0.00907i)11-s + (0.332 − 0.0105i)12-s + (−1.34 + 0.775i)13-s + (1.07 − 0.620i)14-s + (−0.900 − 0.434i)15-s + (0.611 − 1.05i)16-s + 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.898 - 0.439i$
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: \((2,\ 45,\ (\ :2),\ 0.898 - 0.439i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.84805 + 0.427907i\)
\(L(\frac12)\) \(\approx\) \(1.84805 + 0.427907i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.25 - 7.93i)T \)
5 \( 1 + (-1.07 - 24.9i)T \)
good2 \( 1 + (-2.30 + 3.99i)T + (-8 - 13.8i)T^{2} \)
7 \( 1 + (-45.6 - 26.3i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (1.90 + 1.09i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (226. - 131. i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 487.T + 8.35e4T^{2} \)
19 \( 1 - 8.08T + 1.30e5T^{2} \)
23 \( 1 + (178. + 309. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-25.7 - 14.8i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (534. + 925. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.72e3iT - 1.87e6T^{2} \)
41 \( 1 + (-2.23e3 + 1.29e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-597. - 345. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (817. - 1.41e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 3.35e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.24e3 + 719. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (256. - 444. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.80e3 - 2.19e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 4.43e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.04e3iT - 2.83e7T^{2} \)
79 \( 1 + (2.00e3 - 3.47e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (3.25e3 - 5.64e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 7.63e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.10e4 - 6.35e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−14.62178265146248630136468115440, −14.39744570774541449879074187801, −12.29383821184707655613808896764, −11.61597716253910728999962957256, −10.67360345958554769788667608500, −9.652237348901654724203287321301, −7.53818940280239580390097799557, −5.48646590234988132050714496156, −4.07342449869330889311323134070, −2.47276727879630756900254351462, 1.25985597813855205886004460135, 4.87119701299116137496016935983, 5.64056855446151178908517732348, 7.41795278297617756147341144232, 8.009658788046261518648615711234, 10.25322475070549694079732320456, 11.85726727420258004236929707156, 12.85399628595334047908937443351, 13.95302441338562565605885013356, 14.80282455194022798488809758073

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