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Properties

Label 2-825-275.191-c1-0-57
Degree $2$
Conductor $825$
Sign $0.341 - 0.939i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.690 − 2.12i)2-s + 3-s + (−2.42 + 1.76i)4-s + (−1.80 − 1.31i)5-s + (−0.690 − 2.12i)6-s + (0.809 + 2.48i)7-s + (1.80 + 1.31i)8-s + 9-s + (−1.54 + 4.75i)10-s + (0.809 − 3.21i)11-s + (−2.42 + 1.76i)12-s − 6·13-s + (4.73 − 3.44i)14-s + (−1.80 − 1.31i)15-s + (−0.309 + 0.951i)16-s + (1.92 + 5.93i)17-s + ⋯
L(s)  = 1  + (−0.488 − 1.50i)2-s + 0.577·3-s + (−1.21 + 0.881i)4-s + (−0.809 − 0.587i)5-s + (−0.282 − 0.868i)6-s + (0.305 + 0.941i)7-s + (0.639 + 0.464i)8-s + 0.333·9-s + (−0.488 + 1.50i)10-s + (0.243 − 0.969i)11-s + (−0.700 + 0.509i)12-s − 1.66·13-s + (1.26 − 0.919i)14-s + (−0.467 − 0.339i)15-s + (−0.0772 + 0.237i)16-s + (0.467 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.341 - 0.939i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (466, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.341 - 0.939i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + (1.80 + 1.31i)T \)
11 \( 1 + (-0.809 + 3.21i)T \)
good2 \( 1 + (0.690 + 2.12i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (-0.809 - 2.48i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (-1.92 - 5.93i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.35 + 3.88i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (4.42 + 3.21i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (5.73 - 4.16i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (6.04 - 4.39i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.927 + 0.673i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.545 - 1.67i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.94T + 43T^{2} \)
47 \( 1 - 8.09T + 47T^{2} \)
53 \( 1 + (0.309 - 0.951i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.47 + 6.88i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 5.32T + 61T^{2} \)
67 \( 1 + (1.80 - 5.56i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (6.04 + 4.39i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1 - 3.07i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4 - 12.3i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.64 + 8.14i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.88 + 2.09i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.781 + 2.40i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.441198561539617566650852844359, −8.768908644609273748174054668425, −8.421032237593563070837381920028, −7.37350298384482589209268745048, −5.83362192333199917742612595730, −4.55964022260652443507550775659, −3.71653058163608680230468772874, −2.66480727410812985344973786745, −1.72393696323011989326539785248, 0, 2.35647441768233872859518461119, 3.97388040456725791509372221938, 4.66378598874927086404231237800, 5.94236397877721248269784948177, 7.20732095956231384172244018831, 7.51687072516596226603554143489, 7.74196294594757650655064228510, 9.124886774188606158929864251193, 9.754110600473397710551727436630

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