Properties

Label 2-1075-1.1-c5-0-123
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  − 6.87·2-s + 5.75·3-s + 15.2·4-s − 39.5·6-s + 159.·7-s + 115.·8-s − 209.·9-s − 66.6·11-s + 87.7·12-s − 3.15·13-s − 1.09e3·14-s − 1.27e3·16-s + 1.88e3·17-s + 1.44e3·18-s + 1.81e3·19-s + 920.·21-s + 458.·22-s + 2.03e3·23-s + 662.·24-s + 21.6·26-s − 2.60e3·27-s + 2.44e3·28-s − 5.05e3·29-s − 469.·31-s + 5.11e3·32-s − 383.·33-s − 1.29e4·34-s + ⋯
L(s)  = 1  − 1.21·2-s + 0.369·3-s + 0.476·4-s − 0.448·6-s + 1.23·7-s + 0.635·8-s − 0.863·9-s − 0.166·11-s + 0.176·12-s − 0.00516·13-s − 1.49·14-s − 1.24·16-s + 1.58·17-s + 1.04·18-s + 1.15·19-s + 0.455·21-s + 0.201·22-s + 0.802·23-s + 0.234·24-s + 0.00628·26-s − 0.687·27-s + 0.588·28-s − 1.11·29-s − 0.0876·31-s + 0.882·32-s − 0.0613·33-s − 1.92·34-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

\(L(3)\) \(\approx\) \(1.578026987\)
\(L(\frac12)\) \(\approx\) \(1.578026987\)
\(L(\frac{7}{2})\) 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 \( 1 + 6.87T + 32T^{2} \)
3 \( 1 - 5.75T + 243T^{2} \)
7 \( 1 - 159.T + 1.68e4T^{2} \)
11 \( 1 + 66.6T + 1.61e5T^{2} \)
13 \( 1 + 3.15T + 3.71e5T^{2} \)
17 \( 1 - 1.88e3T + 1.41e6T^{2} \)
19 \( 1 - 1.81e3T + 2.47e6T^{2} \)
23 \( 1 - 2.03e3T + 6.43e6T^{2} \)
29 \( 1 + 5.05e3T + 2.05e7T^{2} \)
31 \( 1 + 469.T + 2.86e7T^{2} \)
37 \( 1 - 1.13e4T + 6.93e7T^{2} \)
41 \( 1 + 7.18e3T + 1.15e8T^{2} \)
47 \( 1 - 491.T + 2.29e8T^{2} \)
53 \( 1 - 3.51e4T + 4.18e8T^{2} \)
59 \( 1 - 4.18e4T + 7.14e8T^{2} \)
61 \( 1 - 2.19e4T + 8.44e8T^{2} \)
67 \( 1 + 6.83e4T + 1.35e9T^{2} \)
71 \( 1 + 8.29e3T + 1.80e9T^{2} \)
73 \( 1 - 4.74e4T + 2.07e9T^{2} \)
79 \( 1 - 1.63e4T + 3.07e9T^{2} \)
83 \( 1 + 5.60e3T + 3.93e9T^{2} \)
89 \( 1 - 2.35e4T + 5.58e9T^{2} \)
97 \( 1 - 2.82e4T + 8.58e9T^{2} \)
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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

−9.092815827727739599994166205770, −8.306446241657040054825745327865, −7.79844596656850709069079965478, −7.19948741218049886625758768807, −5.62162292415828923722043155205, −5.04755138781048143210386755001, −3.73360337977605637871403077176, −2.57944792069428019412413649042, −1.47005034758242845904699169901, −0.70108425245302977320723813301, 0.70108425245302977320723813301, 1.47005034758242845904699169901, 2.57944792069428019412413649042, 3.73360337977605637871403077176, 5.04755138781048143210386755001, 5.62162292415828923722043155205, 7.19948741218049886625758768807, 7.79844596656850709069079965478, 8.306446241657040054825745327865, 9.092815827727739599994166205770

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