Properties

Label 2-140-1.1-c5-0-2
Degree $2$
Conductor $140$
Sign $1$
Analytic cond. $22.4537$
Root an. cond. $4.73853$
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  − 20.1·3-s + 25·5-s + 49·7-s + 162.·9-s − 427.·11-s − 646.·13-s − 503.·15-s − 1.02e3·17-s + 2.25e3·19-s − 986.·21-s + 2.59e3·23-s + 625·25-s + 1.62e3·27-s + 869.·29-s + 4.40e3·31-s + 8.59e3·33-s + 1.22e3·35-s + 2.55e3·37-s + 1.30e4·39-s + 6.22e3·41-s + 7.75e3·43-s + 4.05e3·45-s + 1.88e3·47-s + 2.40e3·49-s + 2.07e4·51-s − 1.17e4·53-s − 1.06e4·55-s + ⋯
L(s)  = 1  − 1.29·3-s + 0.447·5-s + 0.377·7-s + 0.666·9-s − 1.06·11-s − 1.06·13-s − 0.577·15-s − 0.863·17-s + 1.43·19-s − 0.487·21-s + 1.02·23-s + 0.200·25-s + 0.430·27-s + 0.192·29-s + 0.823·31-s + 1.37·33-s + 0.169·35-s + 0.306·37-s + 1.37·39-s + 0.578·41-s + 0.639·43-s + 0.298·45-s + 0.124·47-s + 0.142·49-s + 1.11·51-s − 0.573·53-s − 0.475·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(22.4537\)
Root analytic conductor: \(4.73853\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.074286804\)
\(L(\frac12)\) \(\approx\) \(1.074286804\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - 25T \)
7 \( 1 - 49T \)
good3 \( 1 + 20.1T + 243T^{2} \)
11 \( 1 + 427.T + 1.61e5T^{2} \)
13 \( 1 + 646.T + 3.71e5T^{2} \)
17 \( 1 + 1.02e3T + 1.41e6T^{2} \)
19 \( 1 - 2.25e3T + 2.47e6T^{2} \)
23 \( 1 - 2.59e3T + 6.43e6T^{2} \)
29 \( 1 - 869.T + 2.05e7T^{2} \)
31 \( 1 - 4.40e3T + 2.86e7T^{2} \)
37 \( 1 - 2.55e3T + 6.93e7T^{2} \)
41 \( 1 - 6.22e3T + 1.15e8T^{2} \)
43 \( 1 - 7.75e3T + 1.47e8T^{2} \)
47 \( 1 - 1.88e3T + 2.29e8T^{2} \)
53 \( 1 + 1.17e4T + 4.18e8T^{2} \)
59 \( 1 - 3.33e4T + 7.14e8T^{2} \)
61 \( 1 + 1.21e3T + 8.44e8T^{2} \)
67 \( 1 + 5.04e4T + 1.35e9T^{2} \)
71 \( 1 - 5.72e4T + 1.80e9T^{2} \)
73 \( 1 - 8.40e4T + 2.07e9T^{2} \)
79 \( 1 - 9.84e4T + 3.07e9T^{2} \)
83 \( 1 + 3.21e4T + 3.93e9T^{2} \)
89 \( 1 - 5.43e4T + 5.58e9T^{2} \)
97 \( 1 + 2.17e4T + 8.58e9T^{2} \)
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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

−12.11159598582477571529249108955, −11.20921569538552722389922099237, −10.41707015427650963389091043908, −9.347009442680675662037073709182, −7.78948134162268314520418813995, −6.67397800399970079603551464943, −5.40568048479750788843420185290, −4.82714036596842240427396885373, −2.58914842286890635991602355836, −0.72920375490633428883223405250, 0.72920375490633428883223405250, 2.58914842286890635991602355836, 4.82714036596842240427396885373, 5.40568048479750788843420185290, 6.67397800399970079603551464943, 7.78948134162268314520418813995, 9.347009442680675662037073709182, 10.41707015427650963389091043908, 11.20921569538552722389922099237, 12.11159598582477571529249108955

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