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Properties

Label 2-35e2-7.5-c0-0-1
Degree $2$
Conductor $1225$
Sign $0.895 + 0.444i$
Analytic cond. $0.611354$
Root an. cond. $0.781891$
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  + (0.5 − 0.866i)2-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (0.499 + 0.866i)18-s + 0.999·22-s + (0.5 − 0.866i)23-s − 29-s + (0.5 − 0.866i)37-s − 43-s + (−0.499 − 0.866i)46-s + (−1 − 1.73i)53-s + (−0.5 + 0.866i)58-s + 64-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (0.499 + 0.866i)18-s + 0.999·22-s + (0.5 − 0.866i)23-s − 29-s + (0.5 − 0.866i)37-s − 43-s + (−0.499 − 0.866i)46-s + (−1 − 1.73i)53-s + (−0.5 + 0.866i)58-s + 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(0.611354\)
Root analytic conductor: \(0.781891\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ 0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.500352219\)
\(L(\frac12)\) \(\approx\) \(1.500352219\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−10.11273830344097394194785571571, −9.189403386069815823512362787838, −8.183465245407622570640834812019, −7.44912845942094310470123251206, −6.59593430063734229162133716292, −5.30824885253189725655032890565, −4.58160091886499624170858388547, −3.66561247166956056647323094004, −2.57891524588464892588207153204, −1.75470996882189676242365278825, 1.36993470125684026314071655134, 3.09386995085401625795016973948, 4.00807912681082987194580511139, 5.12898918028780398710565239715, 5.93067135229850284971520874692, 6.46455745812204634109356258784, 7.33726212827740936246099960690, 8.212893250609105686544405094365, 9.095461305496279478817919965895, 9.818267397437804726328765840831

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