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Properties

Label 2-825-165.38-c1-0-36
Degree $2$
Conductor $825$
Sign $0.703 - 0.710i$
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  + (−0.520 + 1.02i)2-s + (−0.989 + 1.42i)3-s + (0.404 + 0.556i)4-s + (−0.936 − 1.74i)6-s + (0.785 − 4.96i)7-s + (−3.04 + 0.481i)8-s + (−1.04 − 2.81i)9-s + (2.47 + 2.20i)11-s + (−1.19 + 0.0241i)12-s + (0.692 − 1.35i)13-s + (4.65 + 3.38i)14-s + (0.664 − 2.04i)16-s + (−0.979 + 0.498i)17-s + (3.41 + 0.399i)18-s + (2.83 − 3.90i)19-s + ⋯
L(s)  = 1  + (−0.367 + 0.721i)2-s + (−0.571 + 0.820i)3-s + (0.202 + 0.278i)4-s + (−0.382 − 0.714i)6-s + (0.296 − 1.87i)7-s + (−1.07 + 0.170i)8-s + (−0.347 − 0.937i)9-s + (0.746 + 0.665i)11-s + (−0.343 + 0.00697i)12-s + (0.192 − 0.377i)13-s + (1.24 + 0.903i)14-s + (0.166 − 0.511i)16-s + (−0.237 + 0.120i)17-s + (0.804 + 0.0941i)18-s + (0.651 − 0.896i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.703 - 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03929 + 0.433707i\)
\(L(\frac12)\) \(\approx\) \(1.03929 + 0.433707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.989 - 1.42i)T \)
5 \( 1 \)
11 \( 1 + (-2.47 - 2.20i)T \)
good2 \( 1 + (0.520 - 1.02i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (-0.785 + 4.96i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-0.692 + 1.35i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.979 - 0.498i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-2.83 + 3.90i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.45 + 2.45i)T - 23iT^{2} \)
29 \( 1 + (-6.36 + 4.62i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.04 - 3.20i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.398 - 2.51i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-1.70 + 2.35i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-6.92 - 6.92i)T + 43iT^{2} \)
47 \( 1 + (0.860 + 5.43i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (2.37 + 1.21i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (3.38 - 2.45i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.115 + 0.354i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.65 + 1.65i)T - 67iT^{2} \)
71 \( 1 + (12.2 + 3.97i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.04 - 0.481i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (10.0 - 3.28i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.12 - 13.9i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 16.3T + 89T^{2} \)
97 \( 1 + (-8.19 - 4.17i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.29355354017404623737372718288, −9.522559468837431937920434970049, −8.606043945577900418147596935407, −7.59663366919814035032377291202, −6.88118001172438244580875501618, −6.28363847118113671794200472007, −4.86000895475396320649265812463, −4.16942179291787006070157975888, −3.12051305247610685385137443562, −0.803963788727124928998204713836, 1.20643063298693995713818830155, 2.16175592388440896840956988140, 3.14648417401230149047657625611, 5.05309550109191280738875835325, 5.97612443725903609065260433824, 6.28385885445503777863304465169, 7.60701998510902012731028996400, 8.826145501247897543828088100627, 9.052943586750366150567600938949, 10.28335137574425046034716638831

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