Properties

Label 2-275-1.1-c1-0-10
Degree $2$
Conductor $275$
Sign $-1$
Analytic cond. $2.19588$
Root an. cond. $1.48185$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.30·2-s + 1.30·3-s + 3.30·4-s − 3·6-s − 4.30·7-s − 3.00·8-s − 1.30·9-s − 11-s + 4.30·12-s − 5·13-s + 9.90·14-s + 0.302·16-s + 3.90·17-s + 3.00·18-s − 19-s − 5.60·21-s + 2.30·22-s + 3.69·23-s − 3.90·24-s + 11.5·26-s − 5.60·27-s − 14.2·28-s − 9.90·29-s − 4.21·31-s + 5.30·32-s − 1.30·33-s − 9·34-s + ⋯
L(s)  = 1  − 1.62·2-s + 0.752·3-s + 1.65·4-s − 1.22·6-s − 1.62·7-s − 1.06·8-s − 0.434·9-s − 0.301·11-s + 1.24·12-s − 1.38·13-s + 2.64·14-s + 0.0756·16-s + 0.947·17-s + 0.707·18-s − 0.229·19-s − 1.22·21-s + 0.490·22-s + 0.770·23-s − 0.797·24-s + 2.25·26-s − 1.07·27-s − 2.68·28-s − 1.83·29-s − 0.756·31-s + 0.937·32-s − 0.226·33-s − 1.54·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + 2.30T + 2T^{2} \)
3 \( 1 - 1.30T + 3T^{2} \)
7 \( 1 + 4.30T + 7T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 - 3.90T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 3.69T + 23T^{2} \)
29 \( 1 + 9.90T + 29T^{2} \)
31 \( 1 + 4.21T + 31T^{2} \)
37 \( 1 + 9.60T + 37T^{2} \)
41 \( 1 - 1.60T + 41T^{2} \)
43 \( 1 - 7.21T + 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 2.30T + 53T^{2} \)
59 \( 1 - 0.211T + 59T^{2} \)
61 \( 1 - 2.90T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 4.60T + 71T^{2} \)
73 \( 1 + 2.90T + 73T^{2} \)
79 \( 1 + 0.0916T + 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 - 5.30T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−11.01779809857128900000577135767, −10.00820008617195022368519186425, −9.443182157871460324338093872423, −8.819873036028638500292269708527, −7.61296540418524370081381517166, −7.05460335950302119734931810000, −5.64031011464947950521292302591, −3.39743855585965294744355267594, −2.34628841478842226900834011508, 0, 2.34628841478842226900834011508, 3.39743855585965294744355267594, 5.64031011464947950521292302591, 7.05460335950302119734931810000, 7.61296540418524370081381517166, 8.819873036028638500292269708527, 9.443182157871460324338093872423, 10.00820008617195022368519186425, 11.01779809857128900000577135767

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