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Properties

Label 2-825-11.4-c1-0-10
Degree $2$
Conductor $825$
Sign $0.788 + 0.614i$
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  + (−1.12 − 0.817i)2-s + (−0.309 + 0.951i)3-s + (−0.0207 − 0.0638i)4-s + (1.12 − 0.817i)6-s + (−0.394 − 1.21i)7-s + (−0.888 + 2.73i)8-s + (−0.809 − 0.587i)9-s + (−1.20 + 3.09i)11-s + 0.0671·12-s + (−1.14 − 0.833i)13-s + (−0.548 + 1.68i)14-s + (3.12 − 2.26i)16-s + (4.04 − 2.93i)17-s + (0.429 + 1.32i)18-s + (0.0488 − 0.150i)19-s + ⋯
L(s)  = 1  + (−0.795 − 0.577i)2-s + (−0.178 + 0.549i)3-s + (−0.0103 − 0.0319i)4-s + (0.459 − 0.333i)6-s + (−0.149 − 0.459i)7-s + (−0.313 + 0.966i)8-s + (−0.269 − 0.195i)9-s + (−0.362 + 0.931i)11-s + 0.0193·12-s + (−0.318 − 0.231i)13-s + (−0.146 + 0.451i)14-s + (0.780 − 0.567i)16-s + (0.981 − 0.712i)17-s + (0.101 + 0.311i)18-s + (0.0112 − 0.0345i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.788 + 0.614i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.788 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.765654 - 0.263039i\)
\(L(\frac12)\) \(\approx\) \(0.765654 - 0.263039i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 - 3.09i)T \)
good2 \( 1 + (1.12 + 0.817i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (0.394 + 1.21i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.14 + 0.833i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.04 + 2.93i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.0488 + 0.150i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.00T + 23T^{2} \)
29 \( 1 + (-1.93 - 5.96i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.46 + 1.79i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.45 - 4.46i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.34 + 7.21i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.41T + 43T^{2} \)
47 \( 1 + (-2.54 + 7.82i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.57 - 5.50i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.50 - 7.70i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-11.5 + 8.40i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.38T + 67T^{2} \)
71 \( 1 + (-5.48 + 3.98i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.67 + 8.23i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.05 - 1.49i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.18 + 5.94i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + (-5.18 - 3.76i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.28915360944200148121241083056, −9.468535618896668578091232827704, −8.799857814291111002902251245869, −7.69245886792352146801712451956, −6.88437807563762315816063484724, −5.40557095580294393834597968615, −4.93239756559607382269083861504, −3.50720843450651045119153615217, −2.33679538605375073096955659830, −0.793831081968000064909716372896, 0.880362008215214541359063878965, 2.67208207207784548339927029294, 3.79728187860782538046485939383, 5.35686636213358670313945184102, 6.16440144570047287366593540252, 6.99548349617447977758840391373, 7.912673186049014341910434368970, 8.411037261811816203679822330056, 9.279427683256549803969859129154, 10.08706660176464869373538247876

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