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Properties

Label 2-825-275.136-c1-0-30
Degree $2$
Conductor $825$
Sign $0.506 + 0.862i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
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  − 2.23·2-s + (−0.809 − 0.587i)3-s + 3.00·4-s + (0.690 + 2.12i)5-s + (1.80 + 1.31i)6-s + (0.809 + 2.48i)7-s − 2.23·8-s + (0.309 + 0.951i)9-s + (−1.54 − 4.75i)10-s + (−2.80 − 1.76i)11-s + (−2.42 − 1.76i)12-s + (−1.85 − 5.70i)13-s + (−1.80 − 5.56i)14-s + (0.690 − 2.12i)15-s − 0.999·16-s + (−5.04 − 3.66i)17-s + ⋯
L(s)  = 1  − 1.58·2-s + (−0.467 − 0.339i)3-s + 1.50·4-s + (0.309 + 0.951i)5-s + (0.738 + 0.536i)6-s + (0.305 + 0.941i)7-s − 0.790·8-s + (0.103 + 0.317i)9-s + (−0.488 − 1.50i)10-s + (−0.846 − 0.531i)11-s + (−0.700 − 0.509i)12-s + (−0.514 − 1.58i)13-s + (−0.483 − 1.48i)14-s + (0.178 − 0.549i)15-s − 0.249·16-s + (−1.22 − 0.889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.506 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.388963 - 0.222564i\)
\(L(\frac12)\) \(\approx\) \(0.388963 - 0.222564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.690 - 2.12i)T \)
11 \( 1 + (2.80 + 1.76i)T \)
good2 \( 1 + 2.23T + 2T^{2} \)
7 \( 1 + (-0.809 - 2.48i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.85 + 5.70i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (5.04 + 3.66i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 - 6.61T + 19T^{2} \)
23 \( 1 + (4.42 - 3.21i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 - 7.09T + 29T^{2} \)
31 \( 1 + (-2.30 + 7.10i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.354 + 1.08i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.42 - 1.03i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 7.94T + 43T^{2} \)
47 \( 1 + (6.54 + 4.75i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.61 + 11.1i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.64 - 5.06i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (1.80 + 5.56i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.30 - 7.10i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.61 - 1.90i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-10.4 + 7.60i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.92 - 5.03i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.88 - 2.09i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.04 - 1.48i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.984345030565492572818834860194, −9.434272457348311214385986407182, −8.184185824979360472961167200688, −7.79038810687702875664190511411, −6.91744411806359716055596052847, −5.88899684053943245605554334183, −5.15237705227576180851474008095, −2.89390612638963273856600776176, −2.23908244494546858952363545632, −0.46572899625374888293672892222, 1.07087159161992430178554174663, 2.16322337321163602236814597847, 4.34949359551702993910242742085, 4.82266587045424680147148116961, 6.34745393906952280469454860692, 7.14797210802030311082796579918, 8.013208284328293083962236665509, 8.815728429737669986302288893778, 9.553856435921802883160717076881, 10.21394844446608469412012047783

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