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Properties

Label 2-525-35.12-c1-0-14
Degree $2$
Conductor $525$
Sign $0.686 + 0.726i$
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  + (−1.49 + 0.401i)2-s + (−0.258 + 0.965i)3-s + (0.346 − 0.200i)4-s − 1.54i·6-s + (2.44 + 1.01i)7-s + (1.75 − 1.75i)8-s + (−0.866 − 0.499i)9-s + (−2.59 − 4.49i)11-s + (0.103 + 0.386i)12-s + (−3.30 − 3.30i)13-s + (−4.06 − 0.545i)14-s + (−2.32 + 4.01i)16-s + (−0.0194 − 0.00519i)17-s + (1.49 + 0.401i)18-s + (−1.24 + 2.14i)19-s + ⋯
L(s)  = 1  + (−1.05 + 0.283i)2-s + (−0.149 + 0.557i)3-s + (0.173 − 0.100i)4-s − 0.632i·6-s + (0.922 + 0.385i)7-s + (0.619 − 0.619i)8-s + (−0.288 − 0.166i)9-s + (−0.782 − 1.35i)11-s + (0.0299 + 0.111i)12-s + (−0.917 − 0.917i)13-s + (−1.08 − 0.145i)14-s + (−0.580 + 1.00i)16-s + (−0.00470 − 0.00126i)17-s + (0.352 + 0.0945i)18-s + (−0.284 + 0.492i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.686 + 0.726i$
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: \((2,\ 525,\ (\ :1/2),\ 0.686 + 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.491483 - 0.211813i\)
\(L(\frac12)\) \(\approx\) \(0.491483 - 0.211813i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 - 1.01i)T \)
good2 \( 1 + (1.49 - 0.401i)T + (1.73 - i)T^{2} \)
11 \( 1 + (2.59 + 4.49i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.30 + 3.30i)T + 13iT^{2} \)
17 \( 1 + (0.0194 + 0.00519i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.24 - 2.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.601 + 2.24i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 + (-5.69 + 3.28i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.66 - 0.714i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 3.68iT - 41T^{2} \)
43 \( 1 + (-2.79 + 2.79i)T - 43iT^{2} \)
47 \( 1 + (0.303 + 1.13i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.60 - 1.23i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.222 - 0.385i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.18 - 0.684i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.52 + 5.70i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.14T + 71T^{2} \)
73 \( 1 + (1.91 - 7.15i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.47 - 2.00i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.77 + 3.77i)T + 83iT^{2} \)
89 \( 1 + (1.91 - 3.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.5 + 10.5i)T - 97iT^{2} \)
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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

−10.44404710210510986098168100515, −9.944333865232817577363629327295, −8.793446851168076924289403482351, −8.174187444463453583597610867249, −7.63705069061926193725261046980, −6.09346281588983057465849489627, −5.19799070932434509138246368564, −4.08952155060771065185018174448, −2.54751503813603598060717212197, −0.47919776098583183245682275069, 1.44253921533805736281565581554, 2.37455158207920576528274516585, 4.60312255268827079082815856157, 5.11411905006453116628113832060, 6.95804513374500254751653649543, 7.44226785933917344087374668953, 8.321033898627651996408446280523, 9.222790765722437981658469780237, 10.11825337821876848884975345068, 10.78323931800603817537041453147

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