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Properties

Label 2-825-11.10-c2-0-51
Degree $2$
Conductor $825$
Sign $0.612 - 0.790i$
Analytic cond. $22.4796$
Root an. cond. $4.74126$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.53i·2-s + 1.73·3-s − 8.46·4-s + 6.11i·6-s + 2.58i·7-s − 15.7i·8-s + 2.99·9-s + (6.73 − 8.69i)11-s − 14.6·12-s − 23.7i·13-s − 9.12·14-s + 21.7·16-s − 12.2i·17-s + 10.5i·18-s + 3.27i·19-s + ⋯
L(s)  = 1  + 1.76i·2-s + 0.577·3-s − 2.11·4-s + 1.01i·6-s + 0.369i·7-s − 1.97i·8-s + 0.333·9-s + (0.612 − 0.790i)11-s − 1.22·12-s − 1.82i·13-s − 0.651·14-s + 1.36·16-s − 0.719i·17-s + 0.588i·18-s + 0.172i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(22.4796\)
Root analytic conductor: \(4.74126\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1),\ 0.612 - 0.790i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.809156813\)
\(L(\frac12)\) \(\approx\) \(1.809156813\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
5 \( 1 \)
11 \( 1 + (-6.73 + 8.69i)T \)
good2 \( 1 - 3.53iT - 4T^{2} \)
7 \( 1 - 2.58iT - 49T^{2} \)
13 \( 1 + 23.7iT - 169T^{2} \)
17 \( 1 + 12.2iT - 289T^{2} \)
19 \( 1 - 3.27iT - 361T^{2} \)
23 \( 1 - 14.3T + 529T^{2} \)
29 \( 1 + 38.5iT - 841T^{2} \)
31 \( 1 + 11.1T + 961T^{2} \)
37 \( 1 - 12.5T + 1.36e3T^{2} \)
41 \( 1 + 1.38iT - 1.68e3T^{2} \)
43 \( 1 + 23.9iT - 1.84e3T^{2} \)
47 \( 1 + 19.8T + 2.20e3T^{2} \)
53 \( 1 - 12.0T + 2.80e3T^{2} \)
59 \( 1 + 62.7T + 3.48e3T^{2} \)
61 \( 1 + 21.3iT - 3.72e3T^{2} \)
67 \( 1 - 34T + 4.48e3T^{2} \)
71 \( 1 + 69.2T + 5.04e3T^{2} \)
73 \( 1 - 39.9iT - 5.32e3T^{2} \)
79 \( 1 - 97.6iT - 6.24e3T^{2} \)
83 \( 1 - 71.9iT - 6.88e3T^{2} \)
89 \( 1 - 107.T + 7.92e3T^{2} \)
97 \( 1 - 166.T + 9.40e3T^{2} \)
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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

−9.659150912029623927711335756541, −8.991999946130611811154753340335, −8.196123542329641622785959830228, −7.70608457539777372615023404312, −6.73451365392434235077251288241, −5.81735186363690698635148936785, −5.20994027576057035557404116699, −3.97857235517716099384463709508, −2.85598031219343364047618055633, −0.62095999643772761487767095026, 1.35755137809531496319520345608, 2.05210251186953868394301412097, 3.30543713154342179500215883666, 4.16758637263169974389648538989, 4.76265353752731449974589689272, 6.54186396194847855162332828109, 7.45774262826950513239136507627, 8.941171691212333317481313250182, 9.065256351169939992003892289681, 10.00736673428115949023548775424

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