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Properties

Label 2-525-35.12-c1-0-1
Degree $2$
Conductor $525$
Sign $-0.999 - 0.00550i$
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  + (0.969 − 0.259i)2-s + (−0.258 + 0.965i)3-s + (−0.859 + 0.496i)4-s + 1.00i·6-s + (−2.42 + 1.06i)7-s + (−2.12 + 2.12i)8-s + (−0.866 − 0.499i)9-s + (−1.78 − 3.08i)11-s + (−0.256 − 0.958i)12-s + (−2.78 − 2.78i)13-s + (−2.07 + 1.65i)14-s + (−0.514 + 0.891i)16-s + (0.506 + 0.135i)17-s + (−0.969 − 0.259i)18-s + (−2.06 + 3.57i)19-s + ⋯
L(s)  = 1  + (0.685 − 0.183i)2-s + (−0.149 + 0.557i)3-s + (−0.429 + 0.248i)4-s + 0.409i·6-s + (−0.915 + 0.401i)7-s + (−0.750 + 0.750i)8-s + (−0.288 − 0.166i)9-s + (−0.537 − 0.931i)11-s + (−0.0741 − 0.276i)12-s + (−0.772 − 0.772i)13-s + (−0.554 + 0.443i)14-s + (−0.128 + 0.222i)16-s + (0.122 + 0.0329i)17-s + (−0.228 − 0.0612i)18-s + (−0.473 + 0.819i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.999 - 0.00550i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.999 - 0.00550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000934854 + 0.339742i\)
\(L(\frac12)\) \(\approx\) \(0.000934854 + 0.339742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (2.42 - 1.06i)T \)
good2 \( 1 + (-0.969 + 0.259i)T + (1.73 - i)T^{2} \)
11 \( 1 + (1.78 + 3.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.78 + 2.78i)T + 13iT^{2} \)
17 \( 1 + (-0.506 - 0.135i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.06 - 3.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.668 + 2.49i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 6.14iT - 29T^{2} \)
31 \( 1 + (1.71 - 0.988i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.152 - 0.0409i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 8.28iT - 41T^{2} \)
43 \( 1 + (9.01 - 9.01i)T - 43iT^{2} \)
47 \( 1 + (-1.39 - 5.19i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.55 + 1.48i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.30 - 2.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.67 - 5.00i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.42 + 5.32i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.23T + 71T^{2} \)
73 \( 1 + (-3.98 + 14.8i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (13.1 + 7.57i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.42 - 9.42i)T + 83iT^{2} \)
89 \( 1 + (5.52 - 9.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.48 - 2.48i)T - 97iT^{2} \)
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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

−11.37943076002339716748589988687, −10.39733965508982737330977302288, −9.633715367835886010167496107777, −8.674703262629535833975300581371, −7.933472649490309462695149899458, −6.34402737285930932818402257932, −5.55033570069690439872875509844, −4.71743604738072653301551438951, −3.43657374643157693237872887832, −2.85142781651003567586120575431, 0.15350961579960382466941041685, 2.28751546715259615883127345484, 3.76485781506036277588738038609, 4.75411425447612619796798824608, 5.70395845006841944318527930304, 6.80224555894439770804841354675, 7.26314702734110359872753431316, 8.702642640018645361940479933130, 9.698808043016613251297029976216, 10.18857636083091800153673838015

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