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Properties

Label 2-15e2-5.3-c4-0-3
Degree $2$
Conductor $225$
Sign $-0.850 + 0.525i$
Analytic cond. $23.2582$
Root an. cond. $4.82267$
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  + (−4.58 + 4.58i)2-s − 26i·4-s + (−41.2 + 41.2i)7-s + (45.8 + 45.8i)8-s + 108·11-s + (164. + 164. i)13-s − 378i·14-s − 3.99·16-s + (18.3 − 18.3i)17-s − 140i·19-s + (−494. + 494. i)22-s + (362. + 362. i)23-s − 1.51e3·26-s + (1.07e3 + 1.07e3i)28-s + 810i·29-s + ⋯
L(s)  = 1  + (−1.14 + 1.14i)2-s − 1.62i·4-s + (−0.841 + 0.841i)7-s + (0.716 + 0.716i)8-s + 0.892·11-s + (0.976 + 0.976i)13-s − 1.92i·14-s − 0.0156·16-s + (0.0634 − 0.0634i)17-s − 0.387i·19-s + (−1.02 + 1.02i)22-s + (0.684 + 0.684i)23-s − 2.23·26-s + (1.36 + 1.36i)28-s + 0.963i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(23.2582\)
Root analytic conductor: \(4.82267\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :2),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5458519123\)
\(L(\frac12)\) \(\approx\) \(0.5458519123\)
\(L(3)\) 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 \( 1 \)
good2 \( 1 + (4.58 - 4.58i)T - 16iT^{2} \)
7 \( 1 + (41.2 - 41.2i)T - 2.40e3iT^{2} \)
11 \( 1 - 108T + 1.46e4T^{2} \)
13 \( 1 + (-164. - 164. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-18.3 + 18.3i)T - 8.35e4iT^{2} \)
19 \( 1 + 140iT - 1.30e5T^{2} \)
23 \( 1 + (-362. - 362. i)T + 2.79e5iT^{2} \)
29 \( 1 - 810iT - 7.07e5T^{2} \)
31 \( 1 + 728T + 9.23e5T^{2} \)
37 \( 1 + (659. - 659. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.51e3T + 2.82e6T^{2} \)
43 \( 1 + (41.2 + 41.2i)T + 3.41e6iT^{2} \)
47 \( 1 + (-178. + 178. i)T - 4.87e6iT^{2} \)
53 \( 1 + (3.09e3 + 3.09e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 3.78e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.59e3T + 1.38e7T^{2} \)
67 \( 1 + (3.34e3 - 3.34e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 432T + 2.54e7T^{2} \)
73 \( 1 + (6.43e3 + 6.43e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 8.84e3iT - 3.89e7T^{2} \)
83 \( 1 + (-774. - 774. i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.32e4iT - 6.27e7T^{2} \)
97 \( 1 + (8.08e3 - 8.08e3i)T - 8.85e7iT^{2} \)
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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

−12.02074977556897928633610497164, −10.97937237534761946902447310234, −9.614317667170309339480506075068, −9.118163868920392999246034675434, −8.413193312647999615484114318139, −6.94245977030896287864588006261, −6.48337703064008428697791041750, −5.37265088242829667569315527140, −3.48680625557471128349677838804, −1.43211054334171570286717889394, 0.30890785298436702801199211866, 1.37812183708822884818395354163, 3.06547209903740622525306061957, 3.94083898970673961738253463418, 6.01111647119891484653577755648, 7.25507088558763707392770502541, 8.396119502045717586968755459549, 9.236564027312146186888753306926, 10.19540740247561335273554010055, 10.73641325151875076756697687940

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