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Properties

Label 2-35e2-35.2-c0-0-1
Degree $2$
Conductor $1225$
Sign $-0.156 + 0.987i$
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  + (−0.866 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(0.611354\)
Root analytic conductor: \(0.781891\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (1157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ -0.156 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5147513415\)
\(L(\frac12)\) \(\approx\) \(0.5147513415\)
\(L(1)\) 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

−9.570931391644688875107118037999, −8.674111506046617430614063068744, −8.304570808822917396477936648872, −7.49717488611427114577227235941, −6.05173452693762851578185271129, −5.62122261787881792230381473715, −4.47768030105352114777409982647, −3.42400830224569176083762994909, −2.74191640962392688143513930944, −0.44822589248838267746703118499, 1.76493036919057454849486216818, 2.97780946952674526416349711622, 4.36886402937230402016931794056, 5.06673423761891086002556058371, 5.67138986361282025414797963375, 6.93727355747591879863049198622, 7.76313866342933106516790167616, 8.620009439639750348132438484028, 9.315360781980459496193463487101, 10.24074969274313623380896337861

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