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Properties

Label 2-2804-2804.2803-c0-0-10
Degree $2$
Conductor $2804$
Sign $1$
Analytic cond. $1.39937$
Root an. cond. $1.18295$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.20·3-s + 4-s + 1.47·5-s − 1.20·6-s − 8-s + 0.452·9-s − 1.47·10-s − 0.891·11-s + 1.20·12-s + 1.86·13-s + 1.78·15-s + 16-s − 1.70·17-s − 0.452·18-s + 1.47·20-s + 0.891·22-s + 1.70·23-s − 1.20·24-s + 1.18·25-s − 1.86·26-s − 0.659·27-s − 1.96·29-s − 1.78·30-s − 32-s − 1.07·33-s + 1.70·34-s + ⋯
L(s)  = 1  − 2-s + 1.20·3-s + 4-s + 1.47·5-s − 1.20·6-s − 8-s + 0.452·9-s − 1.47·10-s − 0.891·11-s + 1.20·12-s + 1.86·13-s + 1.78·15-s + 16-s − 1.70·17-s − 0.452·18-s + 1.47·20-s + 0.891·22-s + 1.70·23-s − 1.20·24-s + 1.18·25-s − 1.86·26-s − 0.659·27-s − 1.96·29-s − 1.78·30-s − 32-s − 1.07·33-s + 1.70·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2804\)    =    \(2^{2} \cdot 701\)
Sign: $1$
Analytic conductor: \(1.39937\)
Root analytic conductor: \(1.18295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2804} (2803, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2804,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.528339105\)
\(L(\frac12)\) \(\approx\) \(1.528339105\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
701 \( 1 - T \)
good3 \( 1 - 1.20T + T^{2} \)
5 \( 1 - 1.47T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 0.891T + T^{2} \)
13 \( 1 - 1.86T + T^{2} \)
17 \( 1 + 1.70T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.70T + T^{2} \)
29 \( 1 + 1.96T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.184T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.96T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 0.547T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 0.891T + T^{2} \)
71 \( 1 + 1.47T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 0.184T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 0.547T + T^{2} \)
97 \( 1 + 1.96T + T^{2} \)
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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

−8.903848896907387235678640718334, −8.701520063611185980845545060630, −7.62402412919763515633766411390, −6.91932615501329132952509616374, −5.99976488264558366630080560987, −5.48577433484265969206845061903, −3.92574930310754813633259282280, −2.85408114831330708037109695947, −2.28266759207303809638257782514, −1.43320741742030384736609466220, 1.43320741742030384736609466220, 2.28266759207303809638257782514, 2.85408114831330708037109695947, 3.92574930310754813633259282280, 5.48577433484265969206845061903, 5.99976488264558366630080560987, 6.91932615501329132952509616374, 7.62402412919763515633766411390, 8.701520063611185980845545060630, 8.903848896907387235678640718334

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