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Properties

Label 2-15e2-5.2-c2-0-4
Degree $2$
Conductor $225$
Sign $-0.608 - 0.793i$
Analytic cond. $6.13080$
Root an. cond. $2.47604$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.44 + 2.44i)2-s + 7.99i·4-s + (6.12 + 6.12i)7-s + (−9.79 + 9.79i)8-s − 6·11-s + (3.67 − 3.67i)13-s + 29.9i·14-s − 15.9·16-s + (−17.1 − 17.1i)17-s + 23i·19-s + (−14.6 − 14.6i)22-s + (12.2 − 12.2i)23-s + 18·26-s + (−48.9 + 48.9i)28-s − 6i·29-s + ⋯
L(s)  = 1  + (1.22 + 1.22i)2-s + 1.99i·4-s + (0.874 + 0.874i)7-s + (−1.22 + 1.22i)8-s − 0.545·11-s + (0.282 − 0.282i)13-s + 2.14i·14-s − 0.999·16-s + (−1.00 − 1.00i)17-s + 1.21i·19-s + (−0.668 − 0.668i)22-s + (0.532 − 0.532i)23-s + 0.692·26-s + (−1.74 + 1.74i)28-s − 0.206i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(6.13080\)
Root analytic conductor: \(2.47604\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.31989 + 2.67565i\)
\(L(\frac12)\) \(\approx\) \(1.31989 + 2.67565i\)
\(L(2)\) 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 \( 1 + (-2.44 - 2.44i)T + 4iT^{2} \)
7 \( 1 + (-6.12 - 6.12i)T + 49iT^{2} \)
11 \( 1 + 6T + 121T^{2} \)
13 \( 1 + (-3.67 + 3.67i)T - 169iT^{2} \)
17 \( 1 + (17.1 + 17.1i)T + 289iT^{2} \)
19 \( 1 - 23iT - 361T^{2} \)
23 \( 1 + (-12.2 + 12.2i)T - 529iT^{2} \)
29 \( 1 + 6iT - 841T^{2} \)
31 \( 1 - 25T + 961T^{2} \)
37 \( 1 + (24.4 + 24.4i)T + 1.36e3iT^{2} \)
41 \( 1 - 60T + 1.68e3T^{2} \)
43 \( 1 + (-60.0 + 60.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (7.34 + 7.34i)T + 2.20e3iT^{2} \)
53 \( 1 + (-24.4 + 24.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 18iT - 3.48e3T^{2} \)
61 \( 1 + 37T + 3.72e3T^{2} \)
67 \( 1 + (25.7 + 25.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 132T + 5.04e3T^{2} \)
73 \( 1 + (24.4 - 24.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 10iT - 6.24e3T^{2} \)
83 \( 1 + (2.44 - 2.44i)T - 6.88e3iT^{2} \)
89 \( 1 - 132iT - 7.92e3T^{2} \)
97 \( 1 + (-23.2 - 23.2i)T + 9.40e3iT^{2} \)
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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

−12.54931609004921637395187581488, −11.82854918549371848399205491525, −10.66148781274275470434284595138, −8.958321007283697377248895314114, −8.098513317015904372407622648540, −7.18408848523524831620009154164, −5.94963851198936088537347123490, −5.21695811026559732085479599226, −4.20837470804404606862626668567, −2.59808333354384203725002009685, 1.31513936021459234549442556532, 2.74111739385482804770287422619, 4.20246612758976653161383971469, 4.80162278849052333867531690604, 6.16158494089096047510498295222, 7.56181991852146259686481541420, 8.961731728743597794247281365637, 10.34870494664083020946274555887, 10.98861523980249060461426416409, 11.52888961236261084817115298215

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