Properties

Degree $2$
Conductor $5070$
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 + 5-s + 6-s + 4·7-s + 8-s + 9-s + 10-s + 12-s + 4·14-s + 15-s + 16-s + 6·17-s + 18-s + 4·19-s + 20-s + 4·21-s + 24-s + 25-s + 27-s + 4·28-s − 6·29-s + 30-s − 8·31-s + 32-s + 6·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.872·21-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.755·28-s − 1.11·29-s + 0.182·30-s − 1.43·31-s + 0.176·32-s + 1.02·34-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: $1$
Motivic weight: \(1\)
Character: $\chi_{5070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.577979071\)
\(L(\frac12)\) \(\approx\) \(5.577979071\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.889742230800007366384340287619, −7.70424926324545431494851724983, −6.90106282892871459571015329925, −5.66012343286953819079657082523, −5.41509779315100847068653879171, −4.57317770756600442017159530038, −3.70681432856719735440542242113, −2.95923023255871163375029903372, −1.87988664142882943456143061246, −1.32801143307487185867201245142, 1.32801143307487185867201245142, 1.87988664142882943456143061246, 2.95923023255871163375029903372, 3.70681432856719735440542242113, 4.57317770756600442017159530038, 5.41509779315100847068653879171, 5.66012343286953819079657082523, 6.90106282892871459571015329925, 7.70424926324545431494851724983, 7.889742230800007366384340287619

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