Properties

Degree $2$
Conductor $1225$
Sign $0.813 + 0.581i$
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)4-s + (0.866 + 0.5i)9-s + (−1 − 1.73i)11-s + (0.499 − 0.866i)16-s + 2i·29-s + 0.999·36-s + (−1.73 − 0.999i)44-s − 0.999i·64-s − 2·71-s + (1.73 + i)79-s + (0.499 + 0.866i)81-s − 1.99i·99-s + (−1.73 + i)109-s + (1 + 1.73i)116-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)4-s + (0.866 + 0.5i)9-s + (−1 − 1.73i)11-s + (0.499 − 0.866i)16-s + 2i·29-s + 0.999·36-s + (−1.73 − 0.999i)44-s − 0.999i·64-s − 2·71-s + (1.73 + i)79-s + (0.499 + 0.866i)81-s − 1.99i·99-s + (−1.73 + i)109-s + (1 + 1.73i)116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.813 + 0.581i$
Motivic weight: \(0\)
Character: $\chi_{1225} (618, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ 0.813 + 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.297508695\)
\(L(\frac12)\) \(\approx\) \(1.297508695\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T^{2} \)
3 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - iT^{2} \)
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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

−10.10842161674811469590070569971, −9.010575697299198735540361607344, −8.102313856268269707268245424081, −7.36073443779792664314877648857, −6.52911788623658706797619758861, −5.61815753668855137836644432381, −4.97329414844732307864484567044, −3.49584534378517935283434112741, −2.58310175237063391028245005097, −1.29453476774021505052523383517, 1.80357785514661503823830905434, 2.66488087310503864355433301927, 3.94965068992227505423442553231, 4.73532175608862269840415202864, 5.98991009960892986026453138808, 6.88038970613669655230438498838, 7.50046853287663568076180233908, 8.070986283820880948501133195982, 9.373894503014709617359903289575, 10.06624578909594598452719613275

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