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Properties

Label 2-825-11.9-c1-0-16
Degree $2$
Conductor $825$
Sign $0.119 - 0.992i$
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  + (−0.758 + 2.33i)2-s + (0.809 − 0.587i)3-s + (−3.26 − 2.36i)4-s + (0.758 + 2.33i)6-s + (2.65 + 1.93i)7-s + (4.03 − 2.93i)8-s + (0.309 − 0.951i)9-s + (2.96 − 1.47i)11-s − 4.03·12-s + (0.0967 − 0.297i)13-s + (−6.53 + 4.74i)14-s + (1.29 + 3.98i)16-s + (−1.54 − 4.75i)17-s + (1.98 + 1.44i)18-s + (6.03 − 4.38i)19-s + ⋯
L(s)  = 1  + (−0.536 + 1.65i)2-s + (0.467 − 0.339i)3-s + (−1.63 − 1.18i)4-s + (0.309 + 0.953i)6-s + (1.00 + 0.730i)7-s + (1.42 − 1.03i)8-s + (0.103 − 0.317i)9-s + (0.894 − 0.446i)11-s − 1.16·12-s + (0.0268 − 0.0825i)13-s + (−1.74 + 1.26i)14-s + (0.323 + 0.996i)16-s + (−0.374 − 1.15i)17-s + (0.468 + 0.340i)18-s + (1.38 − 1.00i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.119 - 0.992i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.119 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11201 + 0.985958i\)
\(L(\frac12)\) \(\approx\) \(1.11201 + 0.985958i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 \)
11 \( 1 + (-2.96 + 1.47i)T \)
good2 \( 1 + (0.758 - 2.33i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-2.65 - 1.93i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.0967 + 0.297i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.54 + 4.75i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-6.03 + 4.38i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.07T + 23T^{2} \)
29 \( 1 + (-4.07 - 2.96i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.06 + 3.28i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.13 - 1.54i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (8.77 - 6.37i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.51T + 43T^{2} \)
47 \( 1 + (9.70 - 7.05i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.52 - 4.69i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.41 - 5.38i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.83 - 8.73i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 + (-0.949 - 2.92i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.00 + 5.08i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.67 + 5.14i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.02 + 15.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 1.62T + 89T^{2} \)
97 \( 1 + (0.0692 - 0.213i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.842120359707177454027690034513, −9.137122262945349206263679341220, −8.602199586267953241706119773445, −7.84709506082204646174944087492, −7.09097504292320310443109738545, −6.32043608237596870468989819202, −5.30972858574063794109528851639, −4.60062799814213673826730480155, −2.85452338961852613755180157207, −1.09723895388584241691885316113, 1.26253915589287115210163160930, 2.06301804243813749075509055268, 3.58400679863435834684581048486, 4.03216789647915869915714281821, 5.09482513259426865200507518499, 6.79437767062994301855486597119, 8.091279751608878372685185137924, 8.412079443727871529574195287661, 9.583560125061804981173913035661, 10.02134052368853647165252792858

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