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Properties

Label 2-825-165.38-c1-0-27
Degree $2$
Conductor $825$
Sign $0.343 - 0.939i$
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  + (0.493 − 0.968i)2-s + (0.511 + 1.65i)3-s + (0.481 + 0.662i)4-s + (1.85 + 0.321i)6-s + (−0.654 + 4.13i)7-s + (3.02 − 0.479i)8-s + (−2.47 + 1.69i)9-s + (2.47 − 2.20i)11-s + (−0.850 + 1.13i)12-s + (0.553 − 1.08i)13-s + (3.67 + 2.67i)14-s + (0.522 − 1.60i)16-s + (−4.50 + 2.29i)17-s + (0.416 + 3.23i)18-s + (1.57 − 2.16i)19-s + ⋯
L(s)  = 1  + (0.348 − 0.684i)2-s + (0.295 + 0.955i)3-s + (0.240 + 0.331i)4-s + (0.757 + 0.131i)6-s + (−0.247 + 1.56i)7-s + (1.06 − 0.169i)8-s + (−0.825 + 0.564i)9-s + (0.747 − 0.664i)11-s + (−0.245 + 0.327i)12-s + (0.153 − 0.301i)13-s + (0.982 + 0.713i)14-s + (0.130 − 0.402i)16-s + (−1.09 + 0.556i)17-s + (0.0981 + 0.762i)18-s + (0.360 − 0.495i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.343 - 0.939i$
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: \((2,\ 825,\ (\ :1/2),\ 0.343 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81469 + 1.26868i\)
\(L(\frac12)\) \(\approx\) \(1.81469 + 1.26868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.511 - 1.65i)T \)
5 \( 1 \)
11 \( 1 + (-2.47 + 2.20i)T \)
good2 \( 1 + (-0.493 + 0.968i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (0.654 - 4.13i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-0.553 + 1.08i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (4.50 - 2.29i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-1.57 + 2.16i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.382 + 0.382i)T - 23iT^{2} \)
29 \( 1 + (4.46 - 3.24i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.46 - 7.59i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.212 + 1.34i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-3.50 + 4.82i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-2.31 - 2.31i)T + 43iT^{2} \)
47 \( 1 + (0.934 + 5.90i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (2.96 + 1.50i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-0.878 + 0.638i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.837 + 2.57i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (1.62 - 1.62i)T - 67iT^{2} \)
71 \( 1 + (-15.6 - 5.07i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.99 + 1.10i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-7.14 + 2.31i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.0758 - 0.148i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 8.05T + 89T^{2} \)
97 \( 1 + (-9.85 - 5.02i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.66016719816227655889856036213, −9.471715529380540167798900504024, −8.827491440085694317358534378578, −8.250634503996467635748464355947, −6.82040843471880938428039781833, −5.78038817995377070480148014305, −4.85641067784249415760961177687, −3.73664011028603444399759233784, −3.00639260957093316340694646633, −2.07010232364024307424284308476, 0.978269806705420154340488026533, 2.15282075004321485998731792282, 3.82561766840253493951848979241, 4.65435911681629904403048338472, 6.10960081384863015606619710314, 6.61690931786159988514550259665, 7.40880872398144235628536980098, 7.80019869158130842871437189393, 9.234013471200318253631858165476, 9.952842879151275800016763606481

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