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Properties

Label 2-825-11.3-c1-0-6
Degree $2$
Conductor $825$
Sign $0.112 - 0.993i$
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  + (1.03 − 0.749i)2-s + (−0.309 − 0.951i)3-s + (−0.115 + 0.354i)4-s + (−1.03 − 0.749i)6-s + (−0.810 + 2.49i)7-s + (0.935 + 2.87i)8-s + (−0.809 + 0.587i)9-s + (−3.05 − 1.30i)11-s + 0.372·12-s + (−3.13 + 2.27i)13-s + (1.03 + 3.18i)14-s + (2.52 + 1.83i)16-s + (−4.61 − 3.35i)17-s + (−0.394 + 1.21i)18-s + (2.58 + 7.94i)19-s + ⋯
L(s)  = 1  + (0.729 − 0.530i)2-s + (−0.178 − 0.549i)3-s + (−0.0575 + 0.177i)4-s + (−0.421 − 0.306i)6-s + (−0.306 + 0.942i)7-s + (0.330 + 1.01i)8-s + (−0.269 + 0.195i)9-s + (−0.919 − 0.392i)11-s + 0.107·12-s + (−0.869 + 0.631i)13-s + (0.276 + 0.850i)14-s + (0.630 + 0.457i)16-s + (−1.11 − 0.812i)17-s + (−0.0929 + 0.285i)18-s + (0.592 + 1.82i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.112 - 0.993i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.112 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.847828 + 0.757022i\)
\(L(\frac12)\) \(\approx\) \(0.847828 + 0.757022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 \)
11 \( 1 + (3.05 + 1.30i)T \)
good2 \( 1 + (-1.03 + 0.749i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.810 - 2.49i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.13 - 2.27i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.61 + 3.35i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.58 - 7.94i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.15T + 23T^{2} \)
29 \( 1 + (2.71 - 8.36i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.55 + 4.03i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.13 + 3.47i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.41 + 4.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.75T + 43T^{2} \)
47 \( 1 + (-1.71 - 5.28i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.03 + 5.11i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.96 + 6.04i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-11.0 - 8.03i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 0.241T + 67T^{2} \)
71 \( 1 + (-3.20 - 2.32i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.78 - 8.58i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.60 + 4.79i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.76 + 4.91i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 5.39T + 89T^{2} \)
97 \( 1 + (11.2 - 8.14i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.66522209513632714620756727040, −9.612160155947113193054879518992, −8.635184689349253489684901685178, −7.88672561169826412053078360819, −6.96522417035534806694834394416, −5.71888927289389397559658969024, −5.18399020952312741699556824571, −3.97467876255933443543055345086, −2.76184693150793312604967077913, −2.09145154682307941157144033520, 0.42168635838408105413304537146, 2.63300725607155213203776332259, 4.00291981451320703933477837161, 4.69589498890977645320219175896, 5.40256855805739641973330145658, 6.51793816733708267855917052449, 7.16561186463040864012063941986, 8.155713410298174123561455218857, 9.465869187668662416541837958260, 10.12845674230091057444768918183

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