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Properties

Label 2-825-5.4-c1-0-19
Degree $2$
Conductor $825$
Sign $0.894 + 0.447i$
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  + 1.90i·2-s − i·3-s − 1.62·4-s + 1.90·6-s − 4.42i·7-s + 0.719i·8-s − 9-s + 11-s + 1.62i·12-s + 0.622i·13-s + 8.42·14-s − 4.61·16-s − 5.18i·17-s − 1.90i·18-s − 7.05·19-s + ⋯
L(s)  = 1  + 1.34i·2-s − 0.577i·3-s − 0.811·4-s + 0.776·6-s − 1.67i·7-s + 0.254i·8-s − 0.333·9-s + 0.301·11-s + 0.468i·12-s + 0.172i·13-s + 2.25·14-s − 1.15·16-s − 1.25i·17-s − 0.448i·18-s − 1.61·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22912 - 0.290157i\)
\(L(\frac12)\) \(\approx\) \(1.22912 - 0.290157i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
11 \( 1 - T \)
good2 \( 1 - 1.90iT - 2T^{2} \)
7 \( 1 + 4.42iT - 7T^{2} \)
13 \( 1 - 0.622iT - 13T^{2} \)
17 \( 1 + 5.18iT - 17T^{2} \)
19 \( 1 + 7.05T + 19T^{2} \)
23 \( 1 + 8.85iT - 23T^{2} \)
29 \( 1 - 7.80T + 29T^{2} \)
31 \( 1 - 2.75T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 0.193T + 41T^{2} \)
43 \( 1 + 5.67iT - 43T^{2} \)
47 \( 1 + 2.75iT - 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 - 4.85T + 59T^{2} \)
61 \( 1 - 6.85T + 61T^{2} \)
67 \( 1 + 1.24iT - 67T^{2} \)
71 \( 1 - 2.75T + 71T^{2} \)
73 \( 1 + 4.23iT - 73T^{2} \)
79 \( 1 + 8.56T + 79T^{2} \)
83 \( 1 + 0.133iT - 83T^{2} \)
89 \( 1 + 5.61T + 89T^{2} \)
97 \( 1 - 7.24iT - 97T^{2} \)
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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

−10.21016994992763427718016142471, −8.866396960897427004998479489509, −8.241991683965615809637707023092, −7.32895975383304295692938745382, −6.76421110771962118825082572381, −6.28129337596084193926299431910, −4.84477184131292186806258237260, −4.18583498199407337787606462020, −2.47068811344536406231443898767, −0.61630173451540393108419477284, 1.71452847983268594848801885341, 2.66738533295421041852669406306, 3.62709873265667025876242237144, 4.62968548506691528379060573417, 5.76845360406980463442903253493, 6.55819994393556537648240276035, 8.312011013133390768081120632367, 8.770720173653183306010615550923, 9.739016674954097129569025345905, 10.24197673579170248946118042968

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