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Properties

Label 2-45e2-9.2-c0-0-0
Degree $2$
Conductor $2025$
Sign $0.642 - 0.766i$
Analytic cond. $1.01060$
Root an. cond. $1.00528$
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  + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)7-s + (1 + 1.73i)13-s + (−0.499 + 0.866i)16-s − 19-s + 0.999·28-s + (0.5 + 0.866i)31-s + 37-s + (−0.5 + 0.866i)43-s + (0.999 − 1.73i)52-s + (0.5 − 0.866i)61-s + 0.999·64-s + (1 + 1.73i)67-s + 73-s + (0.5 + 0.866i)76-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)7-s + (1 + 1.73i)13-s + (−0.499 + 0.866i)16-s − 19-s + 0.999·28-s + (0.5 + 0.866i)31-s + 37-s + (−0.5 + 0.866i)43-s + (0.999 − 1.73i)52-s + (0.5 − 0.866i)61-s + 0.999·64-s + (1 + 1.73i)67-s + 73-s + (0.5 + 0.866i)76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8781286482\)
\(L(\frac12)\) \(\approx\) \(0.8781286482\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.331076410058474974286870493666, −8.853595775352230520733664880828, −8.183621047072573277969563578328, −6.60371960462017987041074441834, −6.44922669678100830692383853913, −5.52096187499096349398373524668, −4.57509429385261834517624921323, −3.84532785491247797873940015620, −2.46278380735784161947341682719, −1.43737327027006997425715034767, 0.68572606405478072083835979423, 2.58137558379082117863140717304, 3.56677438690174082918011858111, 4.05015490656107079173933502881, 5.14270186200306609348464876923, 6.13526485963388073650302270338, 6.94196154963907858949298908740, 7.945673564098707855695554147933, 8.206526873808197146681983719185, 9.158141616394196773776233788769

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