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Properties

Label 2-15e2-15.2-c1-0-3
Degree $2$
Conductor $225$
Sign $0.999 - 0.0387i$
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  + (−1.73 + 1.73i)2-s − 3.99i·4-s + (−1.22 − 1.22i)7-s + (3.46 + 3.46i)8-s − 4.24i·11-s + (3.67 − 3.67i)13-s + 4.24·14-s − 3.99·16-s + (1.73 − 1.73i)17-s + 5i·19-s + (7.34 + 7.34i)22-s + (−1.73 − 1.73i)23-s + 12.7i·26-s + (−4.89 + 4.89i)28-s + 4.24·29-s + ⋯
L(s)  = 1  + (−1.22 + 1.22i)2-s − 1.99i·4-s + (−0.462 − 0.462i)7-s + (1.22 + 1.22i)8-s − 1.27i·11-s + (1.01 − 1.01i)13-s + 1.13·14-s − 0.999·16-s + (0.420 − 0.420i)17-s + 1.14i·19-s + (1.56 + 1.56i)22-s + (−0.361 − 0.361i)23-s + 2.49i·26-s + (−0.925 + 0.925i)28-s + 0.787·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.999 - 0.0387i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.999 - 0.0387i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.588700 + 0.0114144i\)
\(L(\frac12)\) \(\approx\) \(0.588700 + 0.0114144i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (1.73 - 1.73i)T - 2iT^{2} \)
7 \( 1 + (1.22 + 1.22i)T + 7iT^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 + (-3.67 + 3.67i)T - 13iT^{2} \)
17 \( 1 + (-1.73 + 1.73i)T - 17iT^{2} \)
19 \( 1 - 5iT - 19T^{2} \)
23 \( 1 + (1.73 + 1.73i)T + 23iT^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + (2.44 + 2.44i)T + 37iT^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 + (-1.22 + 1.22i)T - 43iT^{2} \)
47 \( 1 + (-5.19 + 5.19i)T - 47iT^{2} \)
53 \( 1 + (-6.92 - 6.92i)T + 53iT^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + (3.67 + 3.67i)T + 67iT^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 + (2.44 - 2.44i)T - 73iT^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (1.73 + 1.73i)T + 83iT^{2} \)
89 \( 1 - 8.48T + 89T^{2} \)
97 \( 1 + (-8.57 - 8.57i)T + 97iT^{2} \)
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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

−12.19158687454920965559033486578, −10.64104472361395875133202875396, −10.28914535945330415065062740759, −8.981069512368920473503000174946, −8.287860466477964578511361575132, −7.42431560186497075392247122170, −6.21244013472233448682779907288, −5.61538356234157567298668388837, −3.51662911228896568943635817463, −0.789909441807858349136943020977, 1.62844835723601809902365422192, 2.96186263021755750703819101722, 4.39915894184286617827701242205, 6.36420580567232837920270535802, 7.58641545234287555253528826062, 8.751838628297997093122650332662, 9.402805178380579625629790557277, 10.21165850034485547602030765611, 11.20313618241031853236235548028, 12.00440012228052309141403199162

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