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Properties

Label 2-525-35.12-c1-0-15
Degree $2$
Conductor $525$
Sign $0.657 + 0.753i$
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  + (−0.258 + 0.965i)3-s + (−1.73 + i)4-s + (0.189 − 2.63i)7-s + (−0.866 − 0.499i)9-s + (−3 − 5.19i)11-s + (−0.517 − 1.93i)12-s + (1.41 + 1.41i)13-s + (1.99 − 3.46i)16-s + (5.79 + 1.55i)17-s + (1.73 − 3i)19-s + (2.50 + 0.866i)21-s + (0.707 − 0.707i)27-s + (2.31 + 4.76i)28-s + (−4.5 + 2.59i)31-s + (5.79 − 1.55i)33-s + ⋯
L(s)  = 1  + (−0.149 + 0.557i)3-s + (−0.866 + 0.5i)4-s + (0.0716 − 0.997i)7-s + (−0.288 − 0.166i)9-s + (−0.904 − 1.56i)11-s + (−0.149 − 0.557i)12-s + (0.392 + 0.392i)13-s + (0.499 − 0.866i)16-s + (1.40 + 0.376i)17-s + (0.397 − 0.688i)19-s + (0.545 + 0.188i)21-s + (0.136 − 0.136i)27-s + (0.436 + 0.899i)28-s + (−0.808 + 0.466i)31-s + (1.00 − 0.270i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.657 + 0.753i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.837652 - 0.380707i\)
\(L(\frac12)\) \(\approx\) \(0.837652 - 0.380707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-0.189 + 2.63i)T \)
good2 \( 1 + (1.73 - i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.41 - 1.41i)T + 13iT^{2} \)
17 \( 1 + (-5.79 - 1.55i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.73 + 3i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.36 + 2.24i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + (-3.67 + 3.67i)T - 43iT^{2} \)
47 \( 1 + (1.55 + 5.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (10.0 + 2.68i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.19 + 9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.5 - 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.896 - 3.34i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-0.258 + 0.965i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.33 - 2.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.24 + 4.24i)T + 83iT^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.19 - 9.19i)T - 97iT^{2} \)
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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

−10.72119035529936763936322379444, −9.894394062281860929178178832816, −8.940658782291273901288401336372, −8.150932355695101477601578565263, −7.35172259218114549379334635876, −5.84566665354532054381244768689, −5.03765289579696803967019525471, −3.84656270687051038701794293506, −3.22816677713087200743535681577, −0.61901554323764199639669016041, 1.46398499375201067928077764017, 2.91367530668217284890669431162, 4.58174004981262061270803164047, 5.43919187201726366447868786749, 6.12060755833589788265834748330, 7.67336427236289678577036459303, 8.085582044470746979643493311694, 9.488429414726341352568459379817, 9.803415699825598353392050240605, 10.94619590327040353908772959928

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