Properties

Label 2-1075-1.1-c5-0-121
Degree $2$
Conductor $1075$
Sign $1$
Analytic cond. $172.412$
Root an. cond. $13.1305$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.65·2-s + 3.61·3-s + 26.5·4-s − 27.6·6-s + 122.·7-s + 41.8·8-s − 229.·9-s + 575.·11-s + 95.7·12-s + 428.·13-s − 937.·14-s − 1.16e3·16-s − 302.·17-s + 1.75e3·18-s − 1.24e3·19-s + 442.·21-s − 4.40e3·22-s + 138.·23-s + 151.·24-s − 3.27e3·26-s − 1.70e3·27-s + 3.25e3·28-s + 3.50e3·29-s + 983.·31-s + 7.60e3·32-s + 2.07e3·33-s + 2.31e3·34-s + ⋯
L(s)  = 1  − 1.35·2-s + 0.231·3-s + 0.828·4-s − 0.313·6-s + 0.945·7-s + 0.231·8-s − 0.946·9-s + 1.43·11-s + 0.191·12-s + 0.703·13-s − 1.27·14-s − 1.14·16-s − 0.253·17-s + 1.27·18-s − 0.792·19-s + 0.218·21-s − 1.93·22-s + 0.0547·23-s + 0.0535·24-s − 0.951·26-s − 0.450·27-s + 0.783·28-s + 0.774·29-s + 0.183·31-s + 1.31·32-s + 0.332·33-s + 0.343·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1075\)    =    \(5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(172.412\)
Root analytic conductor: \(13.1305\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1075,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.406145965\)
\(L(\frac12)\) \(\approx\) \(1.406145965\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
43 \( 1 - 1.84e3T \)
good2 \( 1 + 7.65T + 32T^{2} \)
3 \( 1 - 3.61T + 243T^{2} \)
7 \( 1 - 122.T + 1.68e4T^{2} \)
11 \( 1 - 575.T + 1.61e5T^{2} \)
13 \( 1 - 428.T + 3.71e5T^{2} \)
17 \( 1 + 302.T + 1.41e6T^{2} \)
19 \( 1 + 1.24e3T + 2.47e6T^{2} \)
23 \( 1 - 138.T + 6.43e6T^{2} \)
29 \( 1 - 3.50e3T + 2.05e7T^{2} \)
31 \( 1 - 983.T + 2.86e7T^{2} \)
37 \( 1 - 1.29e4T + 6.93e7T^{2} \)
41 \( 1 - 3.64e3T + 1.15e8T^{2} \)
47 \( 1 + 2.44e3T + 2.29e8T^{2} \)
53 \( 1 + 1.67e4T + 4.18e8T^{2} \)
59 \( 1 + 1.31e4T + 7.14e8T^{2} \)
61 \( 1 - 5.69e4T + 8.44e8T^{2} \)
67 \( 1 + 1.51e4T + 1.35e9T^{2} \)
71 \( 1 - 4.79e4T + 1.80e9T^{2} \)
73 \( 1 - 4.29e4T + 2.07e9T^{2} \)
79 \( 1 - 2.07e4T + 3.07e9T^{2} \)
83 \( 1 + 6.54e4T + 3.93e9T^{2} \)
89 \( 1 - 3.29e4T + 5.58e9T^{2} \)
97 \( 1 - 5.16e4T + 8.58e9T^{2} \)
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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

−9.055644961017754822691061485714, −8.356007618584508871709554894955, −8.005822140248231985525384136148, −6.79847847201172240678780150524, −6.09582865342325451129204787887, −4.75340906409981978741047328946, −3.87241359438129254308736879019, −2.45081113079255275431971379691, −1.48578092759358019384451865639, −0.67363234359585576377036777251, 0.67363234359585576377036777251, 1.48578092759358019384451865639, 2.45081113079255275431971379691, 3.87241359438129254308736879019, 4.75340906409981978741047328946, 6.09582865342325451129204787887, 6.79847847201172240678780150524, 8.005822140248231985525384136148, 8.356007618584508871709554894955, 9.055644961017754822691061485714

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