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Properties

Label 2-525-105.59-c1-0-15
Degree $2$
Conductor $525$
Sign $0.502 - 0.864i$
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  + (0.866 + 1.5i)2-s − 1.73·3-s + (−0.5 + 0.866i)4-s + (−1.49 − 2.59i)6-s + (−0.866 − 2.5i)7-s + 1.73·8-s + 2.99·9-s + (3 + 1.73i)11-s + (0.866 − 1.50i)12-s − 3.46·13-s + (3 − 3.46i)14-s + (2.49 + 4.33i)16-s + (5.19 + 3i)17-s + (2.59 + 4.49i)18-s + (6 − 3.46i)19-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s − 1.00·3-s + (−0.250 + 0.433i)4-s + (−0.612 − 1.06i)6-s + (−0.327 − 0.944i)7-s + 0.612·8-s + 0.999·9-s + (0.904 + 0.522i)11-s + (0.249 − 0.433i)12-s − 0.960·13-s + (0.801 − 0.925i)14-s + (0.624 + 1.08i)16-s + (1.26 + 0.727i)17-s + (0.612 + 1.06i)18-s + (1.37 − 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.502 - 0.864i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40843 + 0.809949i\)
\(L(\frac12)\) \(\approx\) \(1.40843 + 0.809949i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
5 \( 1 \)
7 \( 1 + (0.866 + 2.5i)T \)
good2 \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 + (-5.19 - 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6 + 3.46i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.866 - 1.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.73iT - 29T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.46 + 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.2 - 6.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 + (-1.73 + 3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−11.02346201309142116217190310515, −10.11252249528861125368881029891, −9.474953526061651576423257538712, −7.54921814427815681258993273403, −7.31033368435066735433427625140, −6.39972015410778239529421948402, −5.48920790007260281755087379050, −4.66737280527091704175577676502, −3.72082600735245644150677243387, −1.24492861112514253003065757861, 1.23377345390456658824732183758, 2.77490445959488058314570683135, 3.81510172024577813546451272901, 5.08913806255409155908644584269, 5.68339366934249967285302593489, 6.91949507923541747152764644619, 7.904603323672525978875373604773, 9.572473667478549295167904611563, 9.859830998045387455362090201705, 11.11985732582485634054274236918

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