Properties

Degree $2$
Conductor $1225$
Sign $0.602 + 0.797i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (−0.866 + 0.499i)18-s − 0.999i·22-s + (0.866 + 0.5i)23-s + 29-s + (−0.866 − 0.5i)37-s i·43-s + (−0.499 − 0.866i)46-s + (1.73 − i)53-s + (−0.866 − 0.5i)58-s − 64-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (−0.866 + 0.499i)18-s − 0.999i·22-s + (0.866 + 0.5i)23-s + 29-s + (−0.866 − 0.5i)37-s i·43-s + (−0.499 − 0.866i)46-s + (1.73 − i)53-s + (−0.866 − 0.5i)58-s − 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.602 + 0.797i$
Motivic weight: \(0\)
Character: $\chi_{1225} (999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ 0.602 + 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6904748202\)
\(L(\frac12)\) \(\approx\) \(0.6904748202\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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

−9.858218231977292076471550120816, −9.074730006505273925106632902293, −8.551994070807265698245022453178, −7.35417132820602436166106735253, −6.72996770311290153799569698002, −5.56296872112346410891689121620, −4.61555643021546928265907994248, −3.53465378423429131960271617445, −2.18037727613486672710851906101, −1.07470219746686778742392255655, 1.19080637093051140081446699663, 2.85617895297746116276147883246, 4.00195891924855911588036356450, 4.95337304366455609734181093077, 6.15832901537976846558656363582, 6.94733325653433445564612249417, 7.66611400817671502287097488321, 8.536869027870009288013885504184, 8.925033098753727908515378092148, 9.997958086047954778299508009591

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