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Properties

Label 2-2695-2695.989-c0-0-3
Degree $2$
Conductor $2695$
Sign $-0.999 - 0.0427i$
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  + (−1.46 − 1.35i)2-s + (0.222 + 2.96i)4-s + (−0.365 − 0.930i)5-s + (0.974 − 0.222i)7-s + (2.45 − 3.08i)8-s + (0.955 + 0.294i)9-s + (−0.728 + 1.85i)10-s + (−0.955 + 0.294i)11-s + (0.131 − 0.574i)13-s + (−1.72 − 0.997i)14-s + (−4.83 + 0.728i)16-s + (−1.29 − 0.880i)17-s + (−0.997 − 1.72i)18-s + (2.68 − 1.29i)20-s + (1.79 + 0.865i)22-s + ⋯
L(s)  = 1  + (−1.46 − 1.35i)2-s + (0.222 + 2.96i)4-s + (−0.365 − 0.930i)5-s + (0.974 − 0.222i)7-s + (2.45 − 3.08i)8-s + (0.955 + 0.294i)9-s + (−0.728 + 1.85i)10-s + (−0.955 + 0.294i)11-s + (0.131 − 0.574i)13-s + (−1.72 − 0.997i)14-s + (−4.83 + 0.728i)16-s + (−1.29 − 0.880i)17-s + (−0.997 − 1.72i)18-s + (2.68 − 1.29i)20-s + (1.79 + 0.865i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $-0.999 - 0.0427i$
Analytic conductor: \(1.34498\)
Root analytic conductor: \(1.15973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2695} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :0),\ -0.999 - 0.0427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4608599522\)
\(L(\frac12)\) \(\approx\) \(0.4608599522\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.365 + 0.930i)T \)
7 \( 1 + (-0.974 + 0.222i)T \)
11 \( 1 + (0.955 - 0.294i)T \)
good2 \( 1 + (1.46 + 1.35i)T + (0.0747 + 0.997i)T^{2} \)
3 \( 1 + (-0.955 - 0.294i)T^{2} \)
13 \( 1 + (-0.131 + 0.574i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + (1.29 + 0.880i)T + (0.365 + 0.930i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.365 + 0.930i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.848 + 1.06i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.0747 - 0.997i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2} \)
61 \( 1 + (0.988 + 0.149i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.134 + 0.0648i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.218 - 0.202i)T + (0.0747 - 0.997i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.385 + 1.68i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-1.40 - 0.432i)T + (0.826 + 0.563i)T^{2} \)
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

−8.717773005538867618088674222005, −8.078575599378339396369175324636, −7.59132520341507262272280744509, −6.97986014348315848697568218591, −5.05189737490408896542096342273, −4.48927545386794456644851242116, −3.66163276322341407091501170582, −2.34782010463142967799150504031, −1.70514066582846541918155187286, −0.50055968809830891596922625533, 1.46450815457367865343857866523, 2.35325555972764351483042499730, 4.20238872736099066646764394851, 4.99280711236762921201177810659, 5.96143033302811143258588506103, 6.67467110769857406111841859948, 7.20631311632882000695368066452, 7.904120954415321436256724971740, 8.475335364938012260808306453224, 9.130272294774618139451082753040

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