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Properties

Label 2-2695-2695.1374-c0-0-3
Degree $2$
Conductor $2695$
Sign $0.536 + 0.843i$
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  + (1.07 + 0.332i)2-s + (0.222 + 0.151i)4-s + (0.988 − 0.149i)5-s + (−0.974 − 0.222i)7-s + (−0.513 − 0.643i)8-s + (−0.733 − 0.680i)9-s + (1.11 + 0.167i)10-s + (0.733 − 0.680i)11-s + (−0.302 − 1.32i)13-s + (−0.975 − 0.563i)14-s + (−0.437 − 1.11i)16-s + (0.116 + 1.55i)17-s + (−0.563 − 0.975i)18-s + (0.242 + 0.116i)20-s + (1.01 − 0.488i)22-s + ⋯
L(s)  = 1  + (1.07 + 0.332i)2-s + (0.222 + 0.151i)4-s + (0.988 − 0.149i)5-s + (−0.974 − 0.222i)7-s + (−0.513 − 0.643i)8-s + (−0.733 − 0.680i)9-s + (1.11 + 0.167i)10-s + (0.733 − 0.680i)11-s + (−0.302 − 1.32i)13-s + (−0.975 − 0.563i)14-s + (−0.437 − 1.11i)16-s + (0.116 + 1.55i)17-s + (−0.563 − 0.975i)18-s + (0.242 + 0.116i)20-s + (1.01 − 0.488i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $0.536 + 0.843i$
Analytic conductor: \(1.34498\)
Root analytic conductor: \(1.15973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2695} (1374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :0),\ 0.536 + 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.876125906\)
\(L(\frac12)\) \(\approx\) \(1.876125906\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.988 + 0.149i)T \)
7 \( 1 + (0.974 + 0.222i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
good2 \( 1 + (-1.07 - 0.332i)T + (0.826 + 0.563i)T^{2} \)
3 \( 1 + (0.733 + 0.680i)T^{2} \)
13 \( 1 + (0.302 + 1.32i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.116 - 1.55i)T + (-0.988 + 0.149i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.988 + 0.149i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.365 + 0.930i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.367 + 0.460i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.826 - 0.563i)T^{2} \)
53 \( 1 + (-0.365 - 0.930i)T^{2} \)
59 \( 1 + (-1.44 - 0.218i)T + (0.955 + 0.294i)T^{2} \)
61 \( 1 + (-0.365 + 0.930i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (1.48 - 0.716i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-1.77 + 0.548i)T + (0.826 - 0.563i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.385 - 1.68i)T + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (-1.40 - 1.29i)T + (0.0747 + 0.997i)T^{2} \)
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

−8.996653016515818986809688953897, −8.223121128574608197223042529396, −6.93908371099383626938663536189, −6.22886640986566064487790026573, −5.84882364156568518280994526160, −5.33195958692667344134438734336, −3.96727883701053603398769965666, −3.47944362583311279447212326701, −2.59143991588587596621714608358, −0.838596591906768165555204935712, 1.96329888386127398743874775533, 2.63527753926669738862162628560, 3.42987868208756827651263061956, 4.55342673004683964403700181148, 5.13019259890071532171057064473, 5.92176009082171256121099503409, 6.65382429122140804355185750449, 7.29633280743959052512943954319, 8.730321827998392522905369059341, 9.217751581804927843489273624937

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