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Properties

Label 2-525-7.5-c2-0-36
Degree $2$
Conductor $525$
Sign $0.910 + 0.413i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.336 − 0.583i)2-s + (1.5 − 0.866i)3-s + (1.77 + 3.07i)4-s − 1.16i·6-s + (6.82 − 1.55i)7-s + 5.08·8-s + (1.5 − 2.59i)9-s + (0.0223 + 0.0387i)11-s + (5.31 + 3.07i)12-s − 23.0i·13-s + (1.38 − 4.50i)14-s + (−5.38 + 9.32i)16-s + (8.16 − 4.71i)17-s + (−1.01 − 1.74i)18-s + (0.991 + 0.572i)19-s + ⋯
L(s)  = 1  + (0.168 − 0.291i)2-s + (0.5 − 0.288i)3-s + (0.443 + 0.767i)4-s − 0.194i·6-s + (0.974 − 0.222i)7-s + 0.635·8-s + (0.166 − 0.288i)9-s + (0.00203 + 0.00352i)11-s + (0.443 + 0.255i)12-s − 1.76i·13-s + (0.0992 − 0.321i)14-s + (−0.336 + 0.582i)16-s + (0.480 − 0.277i)17-s + (−0.0561 − 0.0972i)18-s + (0.0521 + 0.0301i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.910 + 0.413i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.910 + 0.413i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.964991949\)
\(L(\frac12)\) \(\approx\) \(2.964991949\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-6.82 + 1.55i)T \)
good2 \( 1 + (-0.336 + 0.583i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (-0.0223 - 0.0387i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 23.0iT - 169T^{2} \)
17 \( 1 + (-8.16 + 4.71i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-0.991 - 0.572i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (22.1 - 38.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 53.0T + 841T^{2} \)
31 \( 1 + (-19.5 + 11.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-21.1 + 36.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 38.2iT - 1.68e3T^{2} \)
43 \( 1 + 76.5T + 1.84e3T^{2} \)
47 \( 1 + (-23.5 - 13.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-9.49 - 16.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (4.21 - 2.43i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.6 + 19.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (3.50 + 6.06i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 46.8T + 5.04e3T^{2} \)
73 \( 1 + (72.3 - 41.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (10.2 - 17.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 125. iT - 6.88e3T^{2} \)
89 \( 1 + (-40.4 - 23.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 3.11iT - 9.40e3T^{2} \)
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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

−10.64363952199737689760843181691, −9.882718760922560442268818131866, −8.424124758289013563963882217831, −7.906300507989130635014416998775, −7.35150006003772850875645965575, −5.95579490576441205031807793683, −4.74180408463256597454593792251, −3.53230230708504958243864191889, −2.64966986921072864174717927180, −1.28739852689668380067175168414, 1.47447484808991717827002218735, 2.49123339022612096553975282553, 4.31679819756314617416688229937, 4.88208529626829822477797536275, 6.20552425357858412931693366851, 6.92799253845315665386867876380, 8.147531743681265644445280017499, 8.800266366665805572398254400754, 9.994020759505191763254068997118, 10.52483820629963681667620594448

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