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Properties

Label 2-525-15.2-c1-0-34
Degree $2$
Conductor $525$
Sign $-0.706 + 0.707i$
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  + (1.78 − 1.78i)2-s + (1.47 − 0.912i)3-s − 4.37i·4-s + (1 − 4.25i)6-s + (−0.707 − 0.707i)7-s + (−4.23 − 4.23i)8-s + (1.33 − 2.68i)9-s + 5.84i·11-s + (−3.98 − 6.43i)12-s + (−2.38 + 2.38i)13-s − 2.52·14-s − 6.37·16-s + (3.00 − 3.00i)17-s + (−2.41 − 7.17i)18-s + 4i·19-s + ⋯
L(s)  = 1  + (1.26 − 1.26i)2-s + (0.850 − 0.526i)3-s − 2.18i·4-s + (0.408 − 1.73i)6-s + (−0.267 − 0.267i)7-s + (−1.49 − 1.49i)8-s + (0.445 − 0.895i)9-s + 1.76i·11-s + (−1.15 − 1.85i)12-s + (−0.661 + 0.661i)13-s − 0.674·14-s − 1.59·16-s + (0.729 − 0.729i)17-s + (−0.568 − 1.69i)18-s + 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.706 + 0.707i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.706 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30053 - 3.13503i\)
\(L(\frac12)\) \(\approx\) \(1.30053 - 3.13503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.47 + 0.912i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-1.78 + 1.78i)T - 2iT^{2} \)
11 \( 1 - 5.84iT - 11T^{2} \)
13 \( 1 + (2.38 - 2.38i)T - 13iT^{2} \)
17 \( 1 + (-3.00 + 3.00i)T - 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-2.44 - 2.44i)T + 23iT^{2} \)
29 \( 1 - 2.67T + 29T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 + (4.76 + 4.76i)T + 37iT^{2} \)
41 \( 1 + 5.34iT - 41T^{2} \)
43 \( 1 + (-5.65 + 5.65i)T - 43iT^{2} \)
47 \( 1 + (5.45 - 5.45i)T - 47iT^{2} \)
53 \( 1 + (-3.77 - 3.77i)T + 53iT^{2} \)
59 \( 1 + 5.34T + 59T^{2} \)
61 \( 1 + 4.74T + 61T^{2} \)
67 \( 1 + (-0.887 - 0.887i)T + 67iT^{2} \)
71 \( 1 - 8.51iT - 71T^{2} \)
73 \( 1 + (-0.887 + 0.887i)T - 73iT^{2} \)
79 \( 1 - 2.11iT - 79T^{2} \)
83 \( 1 + (-6.93 - 6.93i)T + 83iT^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 + (-2.38 - 2.38i)T + 97iT^{2} \)
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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

−10.58893569852469851466332239481, −9.752787950942108434560824166105, −9.267101254706353645101390793696, −7.53432658606236829004505012143, −6.93236502870977905572414175839, −5.50216006832842478750821836191, −4.42525148171271518235022420943, −3.59185276168562879762743981882, −2.44882674196325897478590399697, −1.56861212953091561751841924763, 2.95575741258431492287746262243, 3.45277393101984317900523476645, 4.73437092045989046128761828747, 5.52713523218474262628825102086, 6.44818462729505343007177154929, 7.56898572487991168851686319071, 8.314634448552832614193751729639, 8.978069328889177631205155955721, 10.27142178402579409678579952882, 11.30454266278627606610039456677

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