Properties

Degree $2$
Conductor $1225$
Sign $0.999 + 0.00342i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.00342i$
Motivic weight: \(0\)
Character: $\chi_{1225} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ 0.999 + 0.00342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.630266111\)
\(L(\frac12)\) \(\approx\) \(1.630266111\)
\(L(1)\) 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

−10.13329446623597806298932814914, −9.077430605455813115827737050822, −8.229691114857010938940363562602, −7.59845686118879749840970484657, −6.37205653261201891536893642438, −5.50593620852497924811778099128, −4.62448274509125449747838576721, −3.85225095112372907557771622793, −2.89538604763926493334569731822, −1.76351664832382944833496101964, 1.34952014963397499355087515702, 3.03678123928940308422767922130, 4.26322602047823058519856745019, 4.54118539395253706201611134673, 5.84666717386432890729971227593, 6.48433447341575731455902893978, 7.11801742361243060241271858594, 8.144329087936616228688256329159, 9.354242340943961260233146548285, 9.757164784518879194895188463418

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