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Properties

Label 2-525-35.3-c1-0-3
Degree $2$
Conductor $525$
Sign $0.142 - 0.989i$
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  + (−0.464 − 0.124i)2-s + (−0.258 − 0.965i)3-s + (−1.53 − 0.884i)4-s + 0.480i·6-s + (−2.38 + 1.13i)7-s + (1.28 + 1.28i)8-s + (−0.866 + 0.499i)9-s + (1.38 − 2.39i)11-s + (−0.457 + 1.70i)12-s + (−0.707 + 0.707i)13-s + (1.25 − 0.231i)14-s + (1.33 + 2.30i)16-s + (−4.28 + 1.14i)17-s + (0.464 − 0.124i)18-s + (0.280 + 0.486i)19-s + ⋯
L(s)  = 1  + (−0.328 − 0.0880i)2-s + (−0.149 − 0.557i)3-s + (−0.765 − 0.442i)4-s + 0.196i·6-s + (−0.902 + 0.430i)7-s + (0.453 + 0.453i)8-s + (−0.288 + 0.166i)9-s + (0.417 − 0.722i)11-s + (−0.132 + 0.493i)12-s + (−0.196 + 0.196i)13-s + (0.334 − 0.0618i)14-s + (0.333 + 0.577i)16-s + (−1.03 + 0.278i)17-s + (0.109 − 0.0293i)18-s + (0.0643 + 0.111i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.142 - 0.989i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.142 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296629 + 0.256883i\)
\(L(\frac12)\) \(\approx\) \(0.296629 + 0.256883i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.38 - 1.13i)T \)
good2 \( 1 + (0.464 + 0.124i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-1.38 + 2.39i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T - 13iT^{2} \)
17 \( 1 + (4.28 - 1.14i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.280 - 0.486i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.08 - 7.77i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.53iT - 29T^{2} \)
31 \( 1 + (-6.97 - 4.02i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.09 - 1.09i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.21iT - 41T^{2} \)
43 \( 1 + (-1.08 - 1.08i)T + 43iT^{2} \)
47 \( 1 + (2.26 - 8.46i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.727 - 0.194i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.09 - 8.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.5 - 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.58 + 9.65i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (2.03 + 7.60i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.05 - 4.65i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.20 - 6.20i)T - 83iT^{2} \)
89 \( 1 + (7.67 + 13.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.41 - 6.41i)T + 97iT^{2} \)
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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

−11.00999848617233332179646619241, −10.07169817037521619025393539095, −9.155475760312240010845077903778, −8.673891546551313349384259544244, −7.50883738884128068687738232923, −6.33929706204644327768913740790, −5.71069036566267739900814238419, −4.44185841100889404992158212083, −3.08855391963945889657424907992, −1.44191811439055449998496605970, 0.27970190908609294829748080097, 2.76813831258500970483589969325, 4.17690078962958234649121257601, 4.56593799104403385811025961414, 6.16645060560348246475420779055, 7.03934994751527769802679535172, 8.134803942761129535286248568563, 9.028139898352806310064464393250, 9.833314099100014149002379586942, 10.24012062406340126891612818859

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