Properties

Label 2-5070-1.1-c1-0-21
Degree $2$
Conductor $5070$
Sign $1$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
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-s + 4-s + 5-s − 6-s − 1.35·7-s + 8-s + 9-s + 10-s − 5.29·11-s − 12-s − 1.35·14-s − 15-s + 16-s − 2.04·17-s + 18-s + 3.24·19-s + 20-s + 1.35·21-s − 5.29·22-s + 1.33·23-s − 24-s + 25-s − 27-s − 1.35·28-s − 5.13·29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.512·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.59·11-s − 0.288·12-s − 0.362·14-s − 0.258·15-s + 0.250·16-s − 0.496·17-s + 0.235·18-s + 0.744·19-s + 0.223·20-s + 0.296·21-s − 1.12·22-s + 0.278·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.256·28-s − 0.953·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.203044149\)
\(L(\frac12)\) \(\approx\) \(2.203044149\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + 1.35T + 7T^{2} \)
11 \( 1 + 5.29T + 11T^{2} \)
17 \( 1 + 2.04T + 17T^{2} \)
19 \( 1 - 3.24T + 19T^{2} \)
23 \( 1 - 1.33T + 23T^{2} \)
29 \( 1 + 5.13T + 29T^{2} \)
31 \( 1 - 2.80T + 31T^{2} \)
37 \( 1 + 2.24T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 4.41T + 43T^{2} \)
47 \( 1 - 4.82T + 47T^{2} \)
53 \( 1 - 14.2T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 3.62T + 61T^{2} \)
67 \( 1 - 9.03T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 0.259T + 79T^{2} \)
83 \( 1 - 3.02T + 83T^{2} \)
89 \( 1 + 5.47T + 89T^{2} \)
97 \( 1 + 5.85T + 97T^{2} \)
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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

−8.018856590583575090081154943004, −7.29840251424032216439573629048, −6.73594369238836365707173734377, −5.67846028066303987411594358665, −5.56530387514592173957019746677, −4.68379685580992027078147465904, −3.81195017545511677045912165084, −2.80445705041291362075024464616, −2.18118109815535274438921371647, −0.72585712268704062390585265377, 0.72585712268704062390585265377, 2.18118109815535274438921371647, 2.80445705041291362075024464616, 3.81195017545511677045912165084, 4.68379685580992027078147465904, 5.56530387514592173957019746677, 5.67846028066303987411594358665, 6.73594369238836365707173734377, 7.29840251424032216439573629048, 8.018856590583575090081154943004

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