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Properties

Label 2-825-165.38-c1-0-44
Degree $2$
Conductor $825$
Sign $0.988 + 0.151i$
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  + (0.440 − 0.863i)2-s + (1.72 + 0.190i)3-s + (0.622 + 0.857i)4-s + (0.922 − 1.40i)6-s + (−0.0473 + 0.298i)7-s + (2.93 − 0.464i)8-s + (2.92 + 0.655i)9-s + (−0.774 + 3.22i)11-s + (0.909 + 1.59i)12-s + (2.68 − 5.27i)13-s + (0.237 + 0.172i)14-s + (0.234 − 0.720i)16-s + (−3.07 + 1.56i)17-s + (1.85 − 2.24i)18-s + (−0.811 + 1.11i)19-s + ⋯
L(s)  = 1  + (0.311 − 0.610i)2-s + (0.993 + 0.109i)3-s + (0.311 + 0.428i)4-s + (0.376 − 0.573i)6-s + (−0.0178 + 0.112i)7-s + (1.03 − 0.164i)8-s + (0.975 + 0.218i)9-s + (−0.233 + 0.972i)11-s + (0.262 + 0.460i)12-s + (0.746 − 1.46i)13-s + (0.0634 + 0.0460i)14-s + (0.0585 − 0.180i)16-s + (−0.746 + 0.380i)17-s + (0.437 − 0.528i)18-s + (−0.186 + 0.256i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.988 + 0.151i$
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: \((2,\ 825,\ (\ :1/2),\ 0.988 + 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.94770 - 0.224719i\)
\(L(\frac12)\) \(\approx\) \(2.94770 - 0.224719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.190i)T \)
5 \( 1 \)
11 \( 1 + (0.774 - 3.22i)T \)
good2 \( 1 + (-0.440 + 0.863i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (0.0473 - 0.298i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-2.68 + 5.27i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (3.07 - 1.56i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (0.811 - 1.11i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.40 - 4.40i)T - 23iT^{2} \)
29 \( 1 + (-1.39 + 1.01i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.40 + 7.39i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.0288 + 0.182i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-2.87 + 3.96i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (1.62 + 1.62i)T + 43iT^{2} \)
47 \( 1 + (0.484 + 3.05i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (2.91 + 1.48i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (2.68 - 1.94i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.59 + 11.0i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (9.94 - 9.94i)T - 67iT^{2} \)
71 \( 1 + (7.73 + 2.51i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-10.1 - 1.60i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (8.06 - 2.62i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.89 - 3.72i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 + 9.17T + 89T^{2} \)
97 \( 1 + (-4.32 - 2.20i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.29374414587647233681396497249, −9.497553660606468149417951931935, −8.332123681417162167360307792373, −7.83182726211911286293504127630, −7.02296909216739317836167261701, −5.70274632329614022429262611496, −4.33209335914352417313983478688, −3.67698124219630631736266766278, −2.63252547838002993807647832029, −1.76107426619682601824372214215, 1.46903413661201025297587718835, 2.63211281505384001759116616793, 3.98124285853216748223855681389, 4.79474917491112247112271791413, 6.18951555149819634627401255950, 6.68564666905406831156739461013, 7.59049937392251041390540296850, 8.591938183698261023613416730471, 9.099735552809604254608098118643, 10.27025818101235587383133301174

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