Properties

Label 2-35e2-5.3-c0-0-2
Degree $2$
Conductor $1225$
Sign $-0.608 + 0.793i$
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  + (1.22 − 1.22i)2-s − 1.99i·4-s + (−1.22 − 1.22i)8-s i·9-s − 11-s − 0.999·16-s + (−1.22 − 1.22i)18-s + (−1.22 + 1.22i)22-s + (1.22 + 1.22i)23-s i·29-s − 1.99·36-s + (−1.22 + 1.22i)37-s + (1.22 + 1.22i)43-s + 1.99i·44-s + 2.99·46-s + ⋯
L(s)  = 1  + (1.22 − 1.22i)2-s − 1.99i·4-s + (−1.22 − 1.22i)8-s i·9-s − 11-s − 0.999·16-s + (−1.22 − 1.22i)18-s + (−1.22 + 1.22i)22-s + (1.22 + 1.22i)23-s i·29-s − 1.99·36-s + (−1.22 + 1.22i)37-s + (1.22 + 1.22i)43-s + 1.99i·44-s + 2.99·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.608 + 0.793i$
Analytic conductor: \(0.611354\)
Root analytic conductor: \(0.781891\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ -0.608 + 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.881170406\)
\(L(\frac12)\) \(\approx\) \(1.881170406\)
\(L(1)\) 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 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
3 \( 1 + iT^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
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

−9.875843711614699068623930413434, −9.214534471159381407180545567801, −8.020049059268566005772745327633, −6.91240748978798720574315366460, −5.89244075884852514837005178803, −5.20173850773963351762683898968, −4.28739158381938449392032060958, −3.33972925965622599452415660909, −2.64888801856488417467285598299, −1.26785764692904594031077648994, 2.37978861687901926380447512796, 3.41331898879863638145969759964, 4.60297917776591991362979888643, 5.14265728771490066099987508288, 5.84404067243340465214747941895, 6.95859991922082585241453399050, 7.43697108886134340945705141768, 8.276299641931899120166096716015, 9.005042311073999745563272096365, 10.52898966185803739289781108359

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