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Properties

Label 2-825-33.32-c1-0-26
Degree $2$
Conductor $825$
Sign $0.991 + 0.132i$
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  + 0.865·2-s + (−1.36 − 1.07i)3-s − 1.25·4-s + (−1.17 − 0.927i)6-s + 0.393i·7-s − 2.81·8-s + (0.701 + 2.91i)9-s + (2.85 − 1.68i)11-s + (1.70 + 1.34i)12-s + 7.00i·13-s + 0.340i·14-s + 0.0664·16-s + 2.48·17-s + (0.607 + 2.52i)18-s − 4.65i·19-s + ⋯
L(s)  = 1  + 0.612·2-s + (−0.785 − 0.618i)3-s − 0.625·4-s + (−0.480 − 0.378i)6-s + 0.148i·7-s − 0.994·8-s + (0.233 + 0.972i)9-s + (0.860 − 0.509i)11-s + (0.491 + 0.387i)12-s + 1.94i·13-s + 0.0909i·14-s + 0.0166·16-s + 0.603·17-s + (0.143 + 0.595i)18-s − 1.06i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.991 + 0.132i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.991 + 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32344 - 0.0879688i\)
\(L(\frac12)\) \(\approx\) \(1.32344 - 0.0879688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.36 + 1.07i)T \)
5 \( 1 \)
11 \( 1 + (-2.85 + 1.68i)T \)
good2 \( 1 - 0.865T + 2T^{2} \)
7 \( 1 - 0.393iT - 7T^{2} \)
13 \( 1 - 7.00iT - 13T^{2} \)
17 \( 1 - 2.48T + 17T^{2} \)
19 \( 1 + 4.65iT - 19T^{2} \)
23 \( 1 + 4.25iT - 23T^{2} \)
29 \( 1 - 4.41T + 29T^{2} \)
31 \( 1 - 5.27T + 31T^{2} \)
37 \( 1 + 0.787T + 37T^{2} \)
41 \( 1 - 8.52T + 41T^{2} \)
43 \( 1 + 5.93iT - 43T^{2} \)
47 \( 1 - 9.78iT - 47T^{2} \)
53 \( 1 - 9.21iT - 53T^{2} \)
59 \( 1 - 7.30iT - 59T^{2} \)
61 \( 1 - 2.74iT - 61T^{2} \)
67 \( 1 + 7.65T + 67T^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + 3.19iT - 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 - 5.92T + 83T^{2} \)
89 \( 1 + 9.75iT - 89T^{2} \)
97 \( 1 + 5.21T + 97T^{2} \)
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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

−10.32664051731672064211110138611, −9.140804648102099407556689111686, −8.738044735774571200765021036015, −7.38885746624571178358745924531, −6.44331893743680522650671534927, −5.94775043085586203037721715137, −4.68723753285527997072154977179, −4.21890423968226103639095871591, −2.63739662632025619769501357927, −0.999551381748418750265985407384, 0.863476762150814875190048309073, 3.23310263351559469624508816377, 3.93104091591982229969052914465, 4.94926655742389325103966331028, 5.63643734061536729194090432761, 6.36162571415586180320518744267, 7.69685127263207130694333205248, 8.615918150680227246325035145060, 9.873438201452841178160786675112, 9.951513752742548668059880131940

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