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Properties

Label 2-525-15.8-c1-0-22
Degree $2$
Conductor $525$
Sign $-0.259 + 0.965i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
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  + (−0.560 − 0.560i)2-s + (−0.0537 + 1.73i)3-s − 1.37i·4-s + (0.999 − 0.939i)6-s + (0.707 − 0.707i)7-s + (−1.88 + 1.88i)8-s + (−2.99 − 0.186i)9-s − 4.10i·11-s + (2.37 + 0.0737i)12-s + (−1.67 − 1.67i)13-s − 0.792·14-s − 0.627·16-s + (0.664 + 0.664i)17-s + (1.57 + 1.78i)18-s − 4i·19-s + ⋯
L(s)  = 1  + (−0.396 − 0.396i)2-s + (−0.0310 + 0.999i)3-s − 0.686i·4-s + (0.408 − 0.383i)6-s + (0.267 − 0.267i)7-s + (−0.667 + 0.667i)8-s + (−0.998 − 0.0620i)9-s − 1.23i·11-s + (0.685 + 0.0212i)12-s + (−0.465 − 0.465i)13-s − 0.211·14-s − 0.156·16-s + (0.161 + 0.161i)17-s + (0.370 + 0.419i)18-s − 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.259 + 0.965i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.259 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.496533 - 0.647811i\)
\(L(\frac12)\) \(\approx\) \(0.496533 - 0.647811i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0537 - 1.73i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (0.560 + 0.560i)T + 2iT^{2} \)
11 \( 1 + 4.10iT - 11T^{2} \)
13 \( 1 + (1.67 + 1.67i)T + 13iT^{2} \)
17 \( 1 + (-0.664 - 0.664i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-2.44 + 2.44i)T - 23iT^{2} \)
29 \( 1 + 5.98T + 29T^{2} \)
31 \( 1 - 4.74T + 31T^{2} \)
37 \( 1 + (3.35 - 3.35i)T - 37iT^{2} \)
41 \( 1 + 11.9iT - 41T^{2} \)
43 \( 1 + (5.65 + 5.65i)T + 43iT^{2} \)
47 \( 1 + (3.11 + 3.11i)T + 47iT^{2} \)
53 \( 1 + (-8.46 + 8.46i)T - 53iT^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 6.74T + 61T^{2} \)
67 \( 1 + (9.01 - 9.01i)T - 67iT^{2} \)
71 \( 1 - 1.87iT - 71T^{2} \)
73 \( 1 + (9.01 + 9.01i)T + 73iT^{2} \)
79 \( 1 - 15.1iT - 79T^{2} \)
83 \( 1 + (11.8 - 11.8i)T - 83iT^{2} \)
89 \( 1 - 3.75T + 89T^{2} \)
97 \( 1 + (-1.67 + 1.67i)T - 97iT^{2} \)
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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

−10.44610808876186145950183047775, −9.998760896912698596968106877355, −8.840686339941863193052625580115, −8.472664345258363897375568010059, −6.88161276440448897203059230570, −5.61661383758852989699045107179, −5.09061667839709495320816832616, −3.70589957880470185286433358980, −2.51682621966810194644698917053, −0.53945064768022886140231762483, 1.76683156632302510585375790796, 3.04045217010861069246250347620, 4.52173275218070634796764395726, 5.84768888356972282777957344328, 6.91845852345778319729731665101, 7.48072477603870397026464872374, 8.210010893067003803501941819033, 9.151890563658440618563862814789, 9.981909518936749084976399406285, 11.52180364520045044321538432512

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