Properties

Label 2-1075-1.1-c5-0-134
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  − 2.14·2-s + 23.1·3-s − 27.4·4-s − 49.6·6-s − 101.·7-s + 127.·8-s + 294.·9-s + 303.·11-s − 635.·12-s + 946.·13-s + 216.·14-s + 604.·16-s − 661.·17-s − 629.·18-s + 1.28e3·19-s − 2.34e3·21-s − 649.·22-s − 963.·23-s + 2.94e3·24-s − 2.02e3·26-s + 1.18e3·27-s + 2.77e3·28-s + 2.73e3·29-s + 2.49e3·31-s − 5.36e3·32-s + 7.02e3·33-s + 1.41e3·34-s + ⋯
L(s)  = 1  − 0.378·2-s + 1.48·3-s − 0.856·4-s − 0.562·6-s − 0.781·7-s + 0.702·8-s + 1.20·9-s + 0.755·11-s − 1.27·12-s + 1.55·13-s + 0.295·14-s + 0.590·16-s − 0.555·17-s − 0.457·18-s + 0.818·19-s − 1.16·21-s − 0.285·22-s − 0.379·23-s + 1.04·24-s − 0.588·26-s + 0.312·27-s + 0.669·28-s + 0.604·29-s + 0.466·31-s − 0.926·32-s + 1.12·33-s + 0.210·34-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

\(L(3)\) \(\approx\) \(2.828307712\)
\(L(\frac12)\) \(\approx\) \(2.828307712\)
\(L(\frac{7}{2})\) 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 \( 1 + 2.14T + 32T^{2} \)
3 \( 1 - 23.1T + 243T^{2} \)
7 \( 1 + 101.T + 1.68e4T^{2} \)
11 \( 1 - 303.T + 1.61e5T^{2} \)
13 \( 1 - 946.T + 3.71e5T^{2} \)
17 \( 1 + 661.T + 1.41e6T^{2} \)
19 \( 1 - 1.28e3T + 2.47e6T^{2} \)
23 \( 1 + 963.T + 6.43e6T^{2} \)
29 \( 1 - 2.73e3T + 2.05e7T^{2} \)
31 \( 1 - 2.49e3T + 2.86e7T^{2} \)
37 \( 1 - 1.67e3T + 6.93e7T^{2} \)
41 \( 1 + 2.22e3T + 1.15e8T^{2} \)
47 \( 1 - 52.2T + 2.29e8T^{2} \)
53 \( 1 - 2.48e4T + 4.18e8T^{2} \)
59 \( 1 + 1.80e4T + 7.14e8T^{2} \)
61 \( 1 + 2.22e4T + 8.44e8T^{2} \)
67 \( 1 + 1.52e4T + 1.35e9T^{2} \)
71 \( 1 - 4.84e4T + 1.80e9T^{2} \)
73 \( 1 - 2.61e4T + 2.07e9T^{2} \)
79 \( 1 + 6.53e4T + 3.07e9T^{2} \)
83 \( 1 + 1.18e4T + 3.93e9T^{2} \)
89 \( 1 - 4.12e4T + 5.58e9T^{2} \)
97 \( 1 - 9.88e4T + 8.58e9T^{2} \)
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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

−9.047007045852230811190154785514, −8.544848328314010982330384409744, −7.87184100783840715049652681612, −6.82862158895918765074987036319, −5.88515702169416887982972872501, −4.46008668368305926534523163730, −3.69078192096604034688709456084, −3.08663134728049984748824603404, −1.71730226726620372992842150650, −0.75066789420808810219582511976, 0.75066789420808810219582511976, 1.71730226726620372992842150650, 3.08663134728049984748824603404, 3.69078192096604034688709456084, 4.46008668368305926534523163730, 5.88515702169416887982972872501, 6.82862158895918765074987036319, 7.87184100783840715049652681612, 8.544848328314010982330384409744, 9.047007045852230811190154785514

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