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Properties

Label 2-15e2-15.8-c3-0-4
Degree $2$
Conductor $225$
Sign $0.876 - 0.481i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 − 1.07i)2-s − 5.70i·4-s + (−7.85 + 7.85i)7-s + (−14.6 + 14.6i)8-s + 33.7i·11-s + (25.7 + 25.7i)13-s + 16.8·14-s − 14.2·16-s + (66.3 + 66.3i)17-s − 128. i·19-s + (36.0 − 36.0i)22-s + (−110. + 110. i)23-s − 55.0i·26-s + (44.8 + 44.8i)28-s + 268.·29-s + ⋯
L(s)  = 1  + (−0.378 − 0.378i)2-s − 0.713i·4-s + (−0.424 + 0.424i)7-s + (−0.648 + 0.648i)8-s + 0.923i·11-s + (0.548 + 0.548i)13-s + 0.321·14-s − 0.222·16-s + (0.946 + 0.946i)17-s − 1.55i·19-s + (0.349 − 0.349i)22-s + (−1.00 + 1.00i)23-s − 0.415i·26-s + (0.302 + 0.302i)28-s + 1.72·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.876 - 0.481i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.876 - 0.481i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.09043 + 0.279839i\)
\(L(\frac12)\) \(\approx\) \(1.09043 + 0.279839i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (1.07 + 1.07i)T + 8iT^{2} \)
7 \( 1 + (7.85 - 7.85i)T - 343iT^{2} \)
11 \( 1 - 33.7iT - 1.33e3T^{2} \)
13 \( 1 + (-25.7 - 25.7i)T + 2.19e3iT^{2} \)
17 \( 1 + (-66.3 - 66.3i)T + 4.91e3iT^{2} \)
19 \( 1 + 128. iT - 6.85e3T^{2} \)
23 \( 1 + (110. - 110. i)T - 1.21e4iT^{2} \)
29 \( 1 - 268.T + 2.43e4T^{2} \)
31 \( 1 + 2.91T + 2.97e4T^{2} \)
37 \( 1 + (-115. + 115. i)T - 5.06e4iT^{2} \)
41 \( 1 - 251. iT - 6.89e4T^{2} \)
43 \( 1 + (-340. - 340. i)T + 7.95e4iT^{2} \)
47 \( 1 + (126. + 126. i)T + 1.03e5iT^{2} \)
53 \( 1 + (254. - 254. i)T - 1.48e5iT^{2} \)
59 \( 1 - 131.T + 2.05e5T^{2} \)
61 \( 1 - 225.T + 2.26e5T^{2} \)
67 \( 1 + (236. - 236. i)T - 3.00e5iT^{2} \)
71 \( 1 - 29.8iT - 3.57e5T^{2} \)
73 \( 1 + (-41.4 - 41.4i)T + 3.89e5iT^{2} \)
79 \( 1 - 450. iT - 4.93e5T^{2} \)
83 \( 1 + (-729. + 729. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.47e3T + 7.04e5T^{2} \)
97 \( 1 + (1.29e3 - 1.29e3i)T - 9.12e5iT^{2} \)
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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

−11.74905013221944656534019920483, −10.83819854939104455242793057657, −9.817847937617309154025852124620, −9.270874620127778882109713321846, −8.091487314854276967326968666109, −6.64805182786749011730290613530, −5.75698522772262222515869020707, −4.44114559419850387668668888850, −2.69099174535724589619887429164, −1.30597208720066058512190188867, 0.58962892944863314930428340967, 3.00207117076940261684614899478, 3.95394075298064640142376706891, 5.75297966431283236216923126930, 6.72144407428515810249265620261, 7.939940158575042730220009657643, 8.450005227566559662371123850443, 9.745837759748209500616882906370, 10.57523431018522976270396602157, 11.93161451843603322810453503985

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