Properties

Label 2-5390-1.1-c1-0-43
Degree $2$
Conductor $5390$
Sign $1$
Analytic cond. $43.0393$
Root an. cond. $6.56043$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3.10·3-s + 4-s − 5-s − 3.10·6-s + 8-s + 6.62·9-s − 10-s + 11-s − 3.10·12-s + 3.62·13-s + 3.10·15-s + 16-s + 4.20·17-s + 6.62·18-s + 8.15·19-s − 20-s + 22-s + 0.897·23-s − 3.10·24-s + 25-s + 3.62·26-s − 11.2·27-s − 7.30·29-s + 3.10·30-s + 3.42·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.79·3-s + 0.5·4-s − 0.447·5-s − 1.26·6-s + 0.353·8-s + 2.20·9-s − 0.316·10-s + 0.301·11-s − 0.895·12-s + 1.00·13-s + 0.801·15-s + 0.250·16-s + 1.01·17-s + 1.56·18-s + 1.87·19-s − 0.223·20-s + 0.213·22-s + 0.187·23-s − 0.633·24-s + 0.200·25-s + 0.711·26-s − 2.16·27-s − 1.35·29-s + 0.566·30-s + 0.614·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5390\)    =    \(2 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(43.0393\)
Root analytic conductor: \(6.56043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5390} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5390,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.844137395\)
\(L(\frac12)\) \(\approx\) \(1.844137395\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + 3.10T + 3T^{2} \)
13 \( 1 - 3.62T + 13T^{2} \)
17 \( 1 - 4.20T + 17T^{2} \)
19 \( 1 - 8.15T + 19T^{2} \)
23 \( 1 - 0.897T + 23T^{2} \)
29 \( 1 + 7.30T + 29T^{2} \)
31 \( 1 - 3.42T + 31T^{2} \)
37 \( 1 + 1.10T + 37T^{2} \)
41 \( 1 + 12.3T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 - 1.15T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 7.25T + 59T^{2} \)
61 \( 1 - 3.15T + 61T^{2} \)
67 \( 1 + 8.41T + 67T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 + 0.205T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 2.20T + 83T^{2} \)
89 \( 1 - 7.45T + 89T^{2} \)
97 \( 1 + 2.25T + 97T^{2} \)
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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

−7.73692440369810984254840606443, −7.28061498262215748668859079187, −6.56229867678872024569683214806, −5.70955218030609286217854748141, −5.51412599991614368378598629590, −4.67049759453664377370675696912, −3.85150223059448113983509250778, −3.20709728320735941646351302520, −1.52900373535412114496886035988, −0.795173964790414516307664886199, 0.795173964790414516307664886199, 1.52900373535412114496886035988, 3.20709728320735941646351302520, 3.85150223059448113983509250778, 4.67049759453664377370675696912, 5.51412599991614368378598629590, 5.70955218030609286217854748141, 6.56229867678872024569683214806, 7.28061498262215748668859079187, 7.73692440369810984254840606443

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