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Properties

Label 2-525-105.74-c0-0-1
Degree $2$
Conductor $525$
Sign $0.992 + 0.123i$
Analytic cond. $0.262009$
Root an. cond. $0.511868$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s + i·13-s + (−0.499 − 0.866i)16-s + (−0.5 − 0.866i)19-s + (−0.499 − 0.866i)21-s + 0.999i·27-s + (−0.866 + 0.499i)28-s + (−1 + 1.73i)31-s + 0.999·36-s + (−0.866 + 0.5i)37-s + (−0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s + i·13-s + (−0.499 − 0.866i)16-s + (−0.5 − 0.866i)19-s + (−0.499 − 0.866i)21-s + 0.999i·27-s + (−0.866 + 0.499i)28-s + (−1 + 1.73i)31-s + 0.999·36-s + (−0.866 + 0.5i)37-s + (−0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.992 + 0.123i$
Analytic conductor: \(0.262009\)
Root analytic conductor: \(0.511868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :0),\ 0.992 + 0.123i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.165045534\)
\(L(\frac12)\) \(\approx\) \(1.165045534\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−10.66248447487446652035460555201, −10.27333742421310656578074255924, −9.275891972020962195351314748355, −8.740018588070978925754892594727, −7.14948455464147816042188569858, −6.78204462194694503729236379369, −5.38568501068442250150551148058, −4.29285374354381864229391367999, −3.14017202034808787869760428142, −1.88230425332256657895875120726, 2.15564969205768814216054890916, 3.12563581516482922057166614727, 3.92913329207248560771198678931, 5.82541780777366235135103980737, 6.67304428559958013983450015879, 7.67425609426367665189068310467, 8.242098140726517504515491650951, 9.190184051499070775999779001434, 10.04060965113083641074934568960, 11.22091098329499437330851776634

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