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Properties

Label 2-15e2-25.14-c1-0-10
Degree $2$
Conductor $225$
Sign $0.898 + 0.439i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.640 + 0.881i)2-s + (0.251 − 0.772i)4-s + (−0.741 − 2.10i)5-s − 3.08i·7-s + (2.91 − 0.947i)8-s + (1.38 − 2.00i)10-s + (−0.929 + 0.674i)11-s + (−2.39 + 3.30i)13-s + (2.72 − 1.97i)14-s + (1.38 + 1.00i)16-s + (4.40 − 1.42i)17-s + (1.84 + 5.67i)19-s + (−1.81 + 0.0428i)20-s + (−1.19 − 0.386i)22-s + (1.36 + 1.88i)23-s + ⋯
L(s)  = 1  + (0.452 + 0.623i)2-s + (0.125 − 0.386i)4-s + (−0.331 − 0.943i)5-s − 1.16i·7-s + (1.03 − 0.334i)8-s + (0.438 − 0.633i)10-s + (−0.280 + 0.203i)11-s + (−0.665 + 0.915i)13-s + (0.727 − 0.528i)14-s + (0.346 + 0.252i)16-s + (1.06 − 0.346i)17-s + (0.423 + 1.30i)19-s + (−0.406 + 0.00958i)20-s + (−0.253 − 0.0824i)22-s + (0.285 + 0.392i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.898 + 0.439i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.898 + 0.439i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49371 - 0.346145i\)
\(L(\frac12)\) \(\approx\) \(1.49371 - 0.346145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.741 + 2.10i)T \)
good2 \( 1 + (-0.640 - 0.881i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 + 3.08iT - 7T^{2} \)
11 \( 1 + (0.929 - 0.674i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.39 - 3.30i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.40 + 1.42i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.84 - 5.67i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.36 - 1.88i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.63 - 5.02i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.182 + 0.560i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.70 + 9.22i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.67 - 5.57i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.42iT - 43T^{2} \)
47 \( 1 + (5.75 + 1.86i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.08 + 1.00i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.57 + 1.87i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (11.1 - 8.07i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.00 - 0.976i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.99 - 6.14i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.23 - 5.83i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.81 + 11.7i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (11.7 - 3.82i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (0.877 - 0.637i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.30 - 1.39i)T + (78.4 + 57.0i)T^{2} \)
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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

−12.36712403668830546260747568465, −11.25580767576886825449106961594, −10.13880334915008454180962142822, −9.379196618499743881261853621003, −7.70109796772849877021824002291, −7.32496839116629586892013795582, −5.84753836944393431332683979574, −4.84464454319134894078974915257, −3.92169567083004631870322158296, −1.32438514902135483215411524867, 2.57617238245312485617083263754, 3.16345626425921894573352409228, 4.78562845453682811483716632969, 6.04595709268846868441066948601, 7.46613058560152618858143117164, 8.152264754304935802873041367676, 9.615052867246983415826922539188, 10.70214732739816604209481936780, 11.48284235868544734324801936065, 12.23411780659686995888564980602

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