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Properties

Label 2-525-7.2-c1-0-9
Degree $2$
Conductor $525$
Sign $-0.550 - 0.834i$
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  + (−1.25 + 2.17i)2-s + (0.5 + 0.866i)3-s + (−2.16 − 3.75i)4-s − 2.51·6-s + (2.29 − 1.31i)7-s + 5.87·8-s + (−0.499 + 0.866i)9-s + (−0.489 − 0.847i)11-s + (2.16 − 3.75i)12-s + 5.14·13-s + (−0.0275 + 6.65i)14-s + (−3.05 + 5.29i)16-s + (2.07 + 3.59i)17-s + (−1.25 − 2.17i)18-s + (1.15 − 1.99i)19-s + ⋯
L(s)  = 1  + (−0.889 + 1.54i)2-s + (0.288 + 0.499i)3-s + (−1.08 − 1.87i)4-s − 1.02·6-s + (0.868 − 0.496i)7-s + 2.07·8-s + (−0.166 + 0.288i)9-s + (−0.147 − 0.255i)11-s + (0.625 − 1.08i)12-s + 1.42·13-s + (−0.00735 + 1.77i)14-s + (−0.763 + 1.32i)16-s + (0.503 + 0.871i)17-s + (−0.296 − 0.513i)18-s + (0.263 − 0.456i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.550 - 0.834i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.550 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.510750 + 0.948454i\)
\(L(\frac12)\) \(\approx\) \(0.510750 + 0.948454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-2.29 + 1.31i)T \)
good2 \( 1 + (1.25 - 2.17i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (0.489 + 0.847i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.14T + 13T^{2} \)
17 \( 1 + (-2.07 - 3.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.15 + 1.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.53 - 4.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.92T + 29T^{2} \)
31 \( 1 + (0.316 + 0.548i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.52 - 7.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.65T + 41T^{2} \)
43 \( 1 - 0.344T + 43T^{2} \)
47 \( 1 + (-2.11 + 3.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.81 + 6.61i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.908 + 1.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.328 - 0.568i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.62 - 8.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.49T + 71T^{2} \)
73 \( 1 + (2.68 + 4.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.44 + 9.42i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.62T + 83T^{2} \)
89 \( 1 + (8.15 - 14.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.53T + 97T^{2} \)
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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

−10.69537425708183046891292578575, −10.11031888956325004965380139913, −9.049535970338823115437285043029, −8.287401277389689377591349523293, −7.88097887147995962987837819361, −6.72288723535507106386322533428, −5.79729191871443198218464984400, −4.89467821677566266297682604846, −3.67774766797308452607230090295, −1.28595969214337638904767522214, 1.07393955965393451456081623848, 2.15318193339328459686687697280, 3.20654246284331696329934936860, 4.41858395234332988938223648089, 5.89958726449148802540001722924, 7.42902211379632093894844996551, 8.305630465947181803904257036594, 8.785100451483328903063860230217, 9.725877183871836592009015279248, 10.71657299398562604368060637869

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