Properties

Degree $2$
Conductor $8034$
Sign $1$
Motivic weight $1$
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 − 3.09·5-s + 6-s + 3.97·7-s − 8-s + 9-s + 3.09·10-s − 2.08·11-s − 12-s − 13-s − 3.97·14-s + 3.09·15-s + 16-s + 3.07·17-s − 18-s − 6.35·19-s − 3.09·20-s − 3.97·21-s + 2.08·22-s + 0.753·23-s + 24-s + 4.57·25-s + 26-s − 27-s + 3.97·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.38·5-s + 0.408·6-s + 1.50·7-s − 0.353·8-s + 0.333·9-s + 0.978·10-s − 0.629·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 0.799·15-s + 0.250·16-s + 0.744·17-s − 0.235·18-s − 1.45·19-s − 0.692·20-s − 0.867·21-s + 0.445·22-s + 0.157·23-s + 0.204·24-s + 0.915·25-s + 0.196·26-s − 0.192·27-s + 0.750·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8034} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6706260245\)
\(L(\frac12)\) \(\approx\) \(0.6706260245\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + T \)
103 \( 1 - T \)
good5 \( 1 + 3.09T + 5T^{2} \)
7 \( 1 - 3.97T + 7T^{2} \)
11 \( 1 + 2.08T + 11T^{2} \)
17 \( 1 - 3.07T + 17T^{2} \)
19 \( 1 + 6.35T + 19T^{2} \)
23 \( 1 - 0.753T + 23T^{2} \)
29 \( 1 + 7.73T + 29T^{2} \)
31 \( 1 - 5.41T + 31T^{2} \)
37 \( 1 - 6.26T + 37T^{2} \)
41 \( 1 + 3.19T + 41T^{2} \)
43 \( 1 - 4.21T + 43T^{2} \)
47 \( 1 + 9.09T + 47T^{2} \)
53 \( 1 - 1.78T + 53T^{2} \)
59 \( 1 - 3.06T + 59T^{2} \)
61 \( 1 + 7.26T + 61T^{2} \)
67 \( 1 + 4.59T + 67T^{2} \)
71 \( 1 - 6.81T + 71T^{2} \)
73 \( 1 + 1.90T + 73T^{2} \)
79 \( 1 - 3.61T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−7.87187992761329509154911037525, −7.48654221722557826290825800629, −6.62549010413530464787743165729, −5.75426386938774728741740912993, −4.92934811108068414213024729643, −4.41748950961285287882128432092, −3.60924200479765694240213647647, −2.46876311475653908375695291857, −1.54619037512751622868281149254, −0.47841399512480290524082160846, 0.47841399512480290524082160846, 1.54619037512751622868281149254, 2.46876311475653908375695291857, 3.60924200479765694240213647647, 4.41748950961285287882128432092, 4.92934811108068414213024729643, 5.75426386938774728741740912993, 6.62549010413530464787743165729, 7.48654221722557826290825800629, 7.87187992761329509154911037525

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