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Properties

Label 2-825-11.3-c1-0-15
Degree $2$
Conductor $825$
Sign $0.911 - 0.410i$
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  + (0.634 − 0.460i)2-s + (−0.309 − 0.951i)3-s + (−0.428 + 1.31i)4-s + (−0.634 − 0.460i)6-s + (0.469 − 1.44i)7-s + (0.820 + 2.52i)8-s + (−0.809 + 0.587i)9-s + (0.438 + 3.28i)11-s + 1.38·12-s + (−0.377 + 0.274i)13-s + (−0.367 − 1.13i)14-s + (−0.557 − 0.404i)16-s + (1.37 + 0.997i)17-s + (−0.242 + 0.745i)18-s + (1.73 + 5.33i)19-s + ⋯
L(s)  = 1  + (0.448 − 0.325i)2-s + (−0.178 − 0.549i)3-s + (−0.214 + 0.658i)4-s + (−0.258 − 0.188i)6-s + (0.177 − 0.545i)7-s + (0.290 + 0.892i)8-s + (−0.269 + 0.195i)9-s + (0.132 + 0.991i)11-s + 0.399·12-s + (−0.104 + 0.0760i)13-s + (−0.0983 − 0.302i)14-s + (−0.139 − 0.101i)16-s + (0.332 + 0.241i)17-s + (−0.0571 + 0.175i)18-s + (0.397 + 1.22i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.911 - 0.410i$
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: \((2,\ 825,\ (\ :1/2),\ 0.911 - 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68980 + 0.362923i\)
\(L(\frac12)\) \(\approx\) \(1.68980 + 0.362923i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.438 - 3.28i)T \)
good2 \( 1 + (-0.634 + 0.460i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-0.469 + 1.44i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.377 - 0.274i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.37 - 0.997i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.73 - 5.33i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 7.68T + 23T^{2} \)
29 \( 1 + (-0.822 + 2.53i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.20 - 2.32i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.945 - 2.90i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.417 + 1.28i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 12.0T + 43T^{2} \)
47 \( 1 + (-0.790 - 2.43i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.48 - 4.71i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.33 + 7.17i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.99 - 6.53i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.89T + 67T^{2} \)
71 \( 1 + (11.8 + 8.57i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.700 - 2.15i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.85 - 7.15i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.53 + 6.92i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.47T + 89T^{2} \)
97 \( 1 + (-11.4 + 8.34i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.47319715211061246218845563205, −9.439949947907456760417137282855, −8.456353166432047264716218308860, −7.52642796495227488920252777645, −7.12151942342812154160789181277, −5.77033832015042214447307944915, −4.76169799119395624946034166740, −3.90847726748370633644333277167, −2.78074802563560715745787514039, −1.47308546999657889857566254550, 0.850672520270803641683542222928, 2.78124034825219937760333278715, 3.94531269414405280509098266749, 5.11795121762019177367140247491, 5.46478127749518126337557901488, 6.48930592138519394410999849015, 7.40147537497100481289397383675, 8.885830513259787556169411728316, 9.131121850638468841646870216202, 10.20165866296519305931591964002

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