Properties

Label 2-8030-1.1-c1-0-150
Degree $2$
Conductor $8030$
Sign $1$
Analytic cond. $64.1198$
Root an. cond. $8.00748$
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 + 3.42·3-s + 4-s + 5-s − 3.42·6-s + 2.68·7-s − 8-s + 8.74·9-s − 10-s − 11-s + 3.42·12-s − 5.61·13-s − 2.68·14-s + 3.42·15-s + 16-s + 6.29·17-s − 8.74·18-s + 2.23·19-s + 20-s + 9.19·21-s + 22-s − 7.18·23-s − 3.42·24-s + 25-s + 5.61·26-s + 19.6·27-s + 2.68·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.97·3-s + 0.5·4-s + 0.447·5-s − 1.39·6-s + 1.01·7-s − 0.353·8-s + 2.91·9-s − 0.316·10-s − 0.301·11-s + 0.989·12-s − 1.55·13-s − 0.717·14-s + 0.884·15-s + 0.250·16-s + 1.52·17-s − 2.06·18-s + 0.512·19-s + 0.223·20-s + 2.00·21-s + 0.213·22-s − 1.49·23-s − 0.699·24-s + 0.200·25-s + 1.10·26-s + 3.78·27-s + 0.507·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8030\)    =    \(2 \cdot 5 \cdot 11 \cdot 73\)
Sign: $1$
Analytic conductor: \(64.1198\)
Root analytic conductor: \(8.00748\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8030,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.251235780\)
\(L(\frac12)\) \(\approx\) \(4.251235780\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 + T \)
73 \( 1 + T \)
good3 \( 1 - 3.42T + 3T^{2} \)
7 \( 1 - 2.68T + 7T^{2} \)
13 \( 1 + 5.61T + 13T^{2} \)
17 \( 1 - 6.29T + 17T^{2} \)
19 \( 1 - 2.23T + 19T^{2} \)
23 \( 1 + 7.18T + 23T^{2} \)
29 \( 1 - 6.59T + 29T^{2} \)
31 \( 1 + 2.97T + 31T^{2} \)
37 \( 1 + 7.39T + 37T^{2} \)
41 \( 1 - 7.44T + 41T^{2} \)
43 \( 1 - 2.76T + 43T^{2} \)
47 \( 1 - 3.02T + 47T^{2} \)
53 \( 1 - 5.79T + 53T^{2} \)
59 \( 1 + 9.03T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 15.4T + 67T^{2} \)
71 \( 1 + 15.9T + 71T^{2} \)
79 \( 1 + 0.450T + 79T^{2} \)
83 \( 1 + 7.40T + 83T^{2} \)
89 \( 1 + 9.29T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−8.023528140309215873746517130635, −7.39898545137320695792084517438, −7.03870739966632181703741958394, −5.72092079525853225842063221113, −4.92858906705733558842197781902, −4.11823043502504668591528906016, −3.19572765636289200410362080885, −2.47707680147178938963551558976, −1.95102959829321255430758189257, −1.10475980067357862224543207354, 1.10475980067357862224543207354, 1.95102959829321255430758189257, 2.47707680147178938963551558976, 3.19572765636289200410362080885, 4.11823043502504668591528906016, 4.92858906705733558842197781902, 5.72092079525853225842063221113, 7.03870739966632181703741958394, 7.39898545137320695792084517438, 8.023528140309215873746517130635

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