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Properties

Label 2-2695-2695.2199-c0-0-1
Degree $2$
Conductor $2695$
Sign $0.718 - 0.695i$
Analytic cond. $1.34498$
Root an. cond. $1.15973$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.277 − 0.347i)2-s + (0.178 + 0.781i)4-s + (−0.900 − 0.433i)5-s + (−0.623 − 0.781i)7-s + (0.722 + 0.347i)8-s + (0.623 + 0.781i)9-s + (−0.400 + 0.193i)10-s + (0.623 − 0.781i)11-s + (−0.777 + 0.974i)13-s − 0.445·14-s + (−0.400 + 0.193i)16-s + (−0.400 + 1.75i)17-s + 0.445·18-s + (0.178 − 0.781i)20-s + (−0.0990 − 0.433i)22-s + ⋯
L(s)  = 1  + (0.277 − 0.347i)2-s + (0.178 + 0.781i)4-s + (−0.900 − 0.433i)5-s + (−0.623 − 0.781i)7-s + (0.722 + 0.347i)8-s + (0.623 + 0.781i)9-s + (−0.400 + 0.193i)10-s + (0.623 − 0.781i)11-s + (−0.777 + 0.974i)13-s − 0.445·14-s + (−0.400 + 0.193i)16-s + (−0.400 + 1.75i)17-s + 0.445·18-s + (0.178 − 0.781i)20-s + (−0.0990 − 0.433i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $0.718 - 0.695i$
Analytic conductor: \(1.34498\)
Root analytic conductor: \(1.15973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2695} (2199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :0),\ 0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.175869060\)
\(L(\frac12)\) \(\approx\) \(1.175869060\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \)
3 \( 1 + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.400 - 1.75i)T + (-0.900 - 0.433i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 - 2T + T^{2} \)
37 \( 1 + (0.900 + 0.433i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.900 + 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1.24 + 1.56i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.927793766886312778189398033560, −8.285603827729393490434546711148, −7.63244985937952854486061467215, −6.96225857761534994001739993773, −6.24797865305195989263023827064, −4.76028654655301021609149678954, −4.15054059571389306922587230181, −3.77767644430200124063903982883, −2.63470544106479380627898913012, −1.40198919508518554901468866461, 0.75108509298949270153101790115, 2.41650450500442904162714123508, 3.21064316013800316981377161928, 4.43964978767609485227714749085, 4.89345220088770286613774120412, 6.02837591343826006609039134046, 6.70737277894111431551110092448, 7.16067552873967790007950905887, 7.931372513926666884041723509223, 9.184120293091039907927906607076

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