Properties

Label 2-275-1.1-c1-0-14
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  + 1.30·2-s − 2.30·3-s − 0.302·4-s − 3·6-s − 0.697·7-s − 3·8-s + 2.30·9-s − 11-s + 0.697·12-s − 5·13-s − 0.908·14-s − 3.30·16-s − 6.90·17-s + 3.00·18-s − 19-s + 1.60·21-s − 1.30·22-s + 7.30·23-s + 6.90·24-s − 6.51·26-s + 1.60·27-s + 0.211·28-s + 0.908·29-s + 10.2·31-s + 1.69·32-s + 2.30·33-s − 9·34-s + ⋯
L(s)  = 1  + 0.921·2-s − 1.32·3-s − 0.151·4-s − 1.22·6-s − 0.263·7-s − 1.06·8-s + 0.767·9-s − 0.301·11-s + 0.201·12-s − 1.38·13-s − 0.242·14-s − 0.825·16-s − 1.67·17-s + 0.707·18-s − 0.229·19-s + 0.350·21-s − 0.277·22-s + 1.52·23-s + 1.41·24-s − 1.27·26-s + 0.308·27-s + 0.0398·28-s + 0.168·29-s + 1.83·31-s + 0.300·32-s + 0.400·33-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 \( 1 - 1.30T + 2T^{2} \)
3 \( 1 + 2.30T + 3T^{2} \)
7 \( 1 + 0.697T + 7T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + 6.90T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 7.30T + 23T^{2} \)
29 \( 1 - 0.908T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 2.39T + 37T^{2} \)
41 \( 1 + 5.60T + 41T^{2} \)
43 \( 1 + 7.21T + 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 - 1.30T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 + 7.90T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 2.60T + 71T^{2} \)
73 \( 1 - 7.90T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 - 3.51T + 83T^{2} \)
89 \( 1 - 1.69T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−11.67020501557625141701675509837, −10.75770470419501364913570977747, −9.714380402320605627927132262200, −8.605904675749065979491153211911, −6.92346215499356070623832759682, −6.24236146899732046281076286266, −4.96521826308231203896098221582, −4.66495966213269380614410498206, −2.85320199008313282448555336254, 0, 2.85320199008313282448555336254, 4.66495966213269380614410498206, 4.96521826308231203896098221582, 6.24236146899732046281076286266, 6.92346215499356070623832759682, 8.605904675749065979491153211911, 9.714380402320605627927132262200, 10.75770470419501364913570977747, 11.67020501557625141701675509837

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