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Properties

Label 2-2523-87.35-c0-0-2
Degree $2$
Conductor $2523$
Sign $0.510 - 0.859i$
Analytic cond. $1.25914$
Root an. cond. $1.12211$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.623 − 0.781i)6-s + (0.222 − 0.974i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.900 − 0.433i)11-s + (0.900 − 0.433i)13-s + (0.623 + 0.781i)14-s + (0.900 − 0.433i)16-s − 17-s + (0.900 − 0.433i)18-s + (0.900 + 0.433i)21-s + (−0.222 + 0.974i)22-s + (0.623 − 0.781i)24-s + (−0.222 − 0.974i)25-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.623 − 0.781i)6-s + (0.222 − 0.974i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.900 − 0.433i)11-s + (0.900 − 0.433i)13-s + (0.623 + 0.781i)14-s + (0.900 − 0.433i)16-s − 17-s + (0.900 − 0.433i)18-s + (0.900 + 0.433i)21-s + (−0.222 + 0.974i)22-s + (0.623 − 0.781i)24-s + (−0.222 − 0.974i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $0.510 - 0.859i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ 0.510 - 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7939154809\)
\(L(\frac12)\) \(\approx\) \(0.7939154809\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.222 - 0.974i)T \)
29 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T + (-0.222 - 0.974i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (-0.222 + 0.974i)T + (-0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2} \)
13 \( 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + (0.900 - 0.433i)T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.222 + 0.974i)T^{2} \)
37 \( 1 + (-0.623 - 0.781i)T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.222 + 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.900 + 0.433i)T^{2} \)
67 \( 1 + (-0.900 - 0.433i)T + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.222 - 0.974i)T^{2} \)
79 \( 1 + (-0.623 - 0.781i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.623 - 0.781i)T + (-0.222 - 0.974i)T^{2} \)
97 \( 1 + (0.900 - 0.433i)T^{2} \)
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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

−9.075290674130316140948668033697, −8.516866141326428388733053870213, −7.84795043138989476207676539733, −6.85977885846635879674009512125, −6.27691224766498971701174091929, −5.55316526519010959079418338599, −4.14087115939527983283314722695, −3.97010207629327529820515278278, −2.78662899477071849093818123149, −0.78766672838909503018676717996, 1.18838779969392990850082374514, 1.98173228874256899409299350788, 2.69511509960339429108581784128, 3.98933783498493783356251050583, 5.27532470345007444996105123508, 6.03733259571589403954651903229, 6.55554403976147136582449216020, 7.52370273865906531685091145504, 8.510537994458242502978476829168, 9.035057650702593410859713023437

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