Properties

Label 2-6930-1.1-c1-0-40
Degree $2$
Conductor $6930$
Sign $1$
Analytic cond. $55.3363$
Root an. cond. $7.43883$
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 + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s + 6.74·13-s − 14-s + 16-s + 6.74·17-s − 20-s + 22-s − 4.74·23-s + 25-s + 6.74·26-s − 28-s − 2·29-s + 32-s + 6.74·34-s + 35-s − 2.74·37-s − 40-s − 6·41-s + 4·43-s + 44-s − 4.74·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 1.87·13-s − 0.267·14-s + 0.250·16-s + 1.63·17-s − 0.223·20-s + 0.213·22-s − 0.989·23-s + 0.200·25-s + 1.32·26-s − 0.188·28-s − 0.371·29-s + 0.176·32-s + 1.15·34-s + 0.169·35-s − 0.451·37-s − 0.158·40-s − 0.937·41-s + 0.609·43-s + 0.150·44-s − 0.699·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6930\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(55.3363\)
Root analytic conductor: \(7.43883\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6930} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.374104137\)
\(L(\frac12)\) \(\approx\) \(3.374104137\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 - 6.74T + 13T^{2} \)
17 \( 1 - 6.74T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4.74T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2.74T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 6.74T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−7.88594916832782213722805779605, −7.21275183054847475942391177650, −6.34376795523346754118643219604, −5.88968874639855336546153818619, −5.21446452060390176500679630656, −4.10121632927673535516053437105, −3.66257760733444569967005570895, −3.09151955668516811345198854114, −1.82301941694625308049499587760, −0.882893546732091835435674859998, 0.882893546732091835435674859998, 1.82301941694625308049499587760, 3.09151955668516811345198854114, 3.66257760733444569967005570895, 4.10121632927673535516053437105, 5.21446452060390176500679630656, 5.88968874639855336546153818619, 6.34376795523346754118643219604, 7.21275183054847475942391177650, 7.88594916832782213722805779605

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