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Properties

Label 2-525-105.59-c1-0-12
Degree $2$
Conductor $525$
Sign $-0.827 - 0.561i$
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  + (0.846 + 1.46i)2-s + (0.0718 + 1.73i)3-s + (−0.433 + 0.750i)4-s + (−2.47 + 1.57i)6-s + (2.01 + 1.71i)7-s + 1.91·8-s + (−2.98 + 0.248i)9-s + (0.399 + 0.230i)11-s + (−1.32 − 0.695i)12-s − 3.38·13-s + (−0.803 + 4.40i)14-s + (2.49 + 4.31i)16-s + (4.76 + 2.75i)17-s + (−2.89 − 4.17i)18-s + (−3.49 + 2.01i)19-s + ⋯
L(s)  = 1  + (0.598 + 1.03i)2-s + (0.0415 + 0.999i)3-s + (−0.216 + 0.375i)4-s + (−1.01 + 0.641i)6-s + (0.762 + 0.647i)7-s + 0.678·8-s + (−0.996 + 0.0829i)9-s + (0.120 + 0.0695i)11-s + (−0.383 − 0.200i)12-s − 0.938·13-s + (−0.214 + 1.17i)14-s + (0.622 + 1.07i)16-s + (1.15 + 0.667i)17-s + (−0.682 − 0.983i)18-s + (−0.801 + 0.462i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.827 - 0.561i$
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: \((2,\ 525,\ (\ :1/2),\ -0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.630407 + 2.05392i\)
\(L(\frac12)\) \(\approx\) \(0.630407 + 2.05392i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0718 - 1.73i)T \)
5 \( 1 \)
7 \( 1 + (-2.01 - 1.71i)T \)
good2 \( 1 + (-0.846 - 1.46i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-0.399 - 0.230i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.38T + 13T^{2} \)
17 \( 1 + (-4.76 - 2.75i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.49 - 2.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.25 + 3.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.71iT - 29T^{2} \)
31 \( 1 + (3.01 + 1.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.02 + 2.89i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.25T + 41T^{2} \)
43 \( 1 - 8.35iT - 43T^{2} \)
47 \( 1 + (2.73 - 1.57i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.78 + 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.88 + 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.90 - 3.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.793 - 0.458i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.52iT - 71T^{2} \)
73 \( 1 + (-3.89 + 6.75i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.58 + 6.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.5iT - 83T^{2} \)
89 \( 1 + (1.35 + 2.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.44T + 97T^{2} \)
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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

−11.13942033906389152753775457273, −10.26160389801929515707769170753, −9.487371379877846424108060192267, −8.198568893030359027017908557217, −7.81086524463625690182251608315, −6.26319539986027786201383345935, −5.64459715297456696169982365519, −4.73228097386467167617053496561, −3.98751510705981340358665466825, −2.27065662335814298993184730853, 1.16000390642004420030823611751, 2.29708810798472896646894360767, 3.39918350027655405734526011258, 4.64633208708881265211546155297, 5.59770245764512946004778390852, 7.25364283699419182017346494979, 7.45716219129912317610420959738, 8.655107566540428551031287027967, 9.932970921689520305419953764866, 10.85490630495989627739785056457

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