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Properties

Label 2-525-35.4-c1-0-23
Degree $2$
Conductor $525$
Sign $0.0667 + 0.997i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0667 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0667 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.0667 + 0.997i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.0667 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.82322 - 2.64066i\)
\(L(\frac12)\) \(\approx\) \(2.82322 - 2.64066i\)
\(L(\frac{3}{2})\) 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

−10.73168487445978096991487473273, −9.976417877846012938509622957119, −9.480243437650430178070400389696, −7.75581853883615541563436407544, −6.54787758089028719087239079926, −5.78489045117012556448294224486, −4.59915379555740688556182183768, −3.72262554755133066376426886682, −3.01221117458627481195230185276, −1.66473060095041804583089927406, 2.70653786707697225791758963890, 3.26795725673336442316719879096, 4.50033114044562373292835902658, 5.60136736160809094786064007969, 6.33709539773140678720821517948, 7.21585775187993724732727289413, 7.989040406545231921082794673294, 8.954040956129891033770845815750, 10.24171912934785516488463426135, 11.63794547245872909103047251429

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