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Properties

Label 2-525-35.9-c1-0-5
Degree $2$
Conductor $525$
Sign $0.758 + 0.652i$
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  + (−2.36 − 1.36i)2-s + (−0.866 + 0.5i)3-s + (2.73 + 4.73i)4-s + 2.73·6-s + (2.5 − 0.866i)7-s − 9.46i·8-s + (0.499 − 0.866i)9-s + (−0.366 − 0.633i)11-s + (−4.73 − 2.73i)12-s + 2.26i·13-s + (−7.09 − 1.36i)14-s + (−7.46 + 12.9i)16-s + (−2.83 + 1.63i)17-s + (−2.36 + 1.36i)18-s + (2.23 − 3.86i)19-s + ⋯
L(s)  = 1  + (−1.67 − 0.965i)2-s + (−0.499 + 0.288i)3-s + (1.36 + 2.36i)4-s + 1.11·6-s + (0.944 − 0.327i)7-s − 3.34i·8-s + (0.166 − 0.288i)9-s + (−0.110 − 0.191i)11-s + (−1.36 − 0.788i)12-s + 0.629i·13-s + (−1.89 − 0.365i)14-s + (−1.86 + 3.23i)16-s + (−0.686 + 0.396i)17-s + (−0.557 + 0.321i)18-s + (0.512 − 0.886i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.758 + 0.652i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.758 + 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.560707 - 0.207953i\)
\(L(\frac12)\) \(\approx\) \(0.560707 - 0.207953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.5 + 0.866i)T \)
good2 \( 1 + (2.36 + 1.36i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (0.366 + 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.26iT - 13T^{2} \)
17 \( 1 + (2.83 - 1.63i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.23 + 3.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.09 - 2.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.19T + 29T^{2} \)
31 \( 1 + (-0.232 - 0.401i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.76 + 1.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.732T + 41T^{2} \)
43 \( 1 - 3.19iT - 43T^{2} \)
47 \( 1 + (-1.73 - i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.7 + 6.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.0980 + 0.169i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.6 + 7.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.19T + 71T^{2} \)
73 \( 1 + (-10.9 + 6.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.69 - 6.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.1iT - 83T^{2} \)
89 \( 1 + (-7.56 + 13.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.9iT - 97T^{2} \)
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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

−10.84586745596834036662060922295, −9.959256888006560635572026712680, −9.072327433823800612693924937161, −8.428791391155842233210599619252, −7.39999107185373102339085595692, −6.64899230199804599491697035486, −4.88605919926622449298858228661, −3.66807239622448054706647150160, −2.24208178343154337055165461172, −0.937870288855792845004158091441, 0.952338313780787739264269641214, 2.26998119543594401573493276492, 4.94380827983789894853145636433, 5.66465800535425033931959532244, 6.69952447536074903986805879981, 7.48052191516048858585644476890, 8.260745719372168323114872624925, 8.926567415381935974093561614181, 10.02291229300123093177135532580, 10.69529060163129191492984808929

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