Properties

Label 2-4334-1.1-c1-0-91
Degree $2$
Conductor $4334$
Sign $1$
Analytic cond. $34.6071$
Root an. cond. $5.88278$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.56·3-s + 4-s − 2.07·5-s + 1.56·6-s + 4.64·7-s + 8-s − 0.555·9-s − 2.07·10-s − 11-s + 1.56·12-s + 6.09·13-s + 4.64·14-s − 3.24·15-s + 16-s + 1.78·17-s − 0.555·18-s + 6.90·19-s − 2.07·20-s + 7.26·21-s − 22-s − 5.48·23-s + 1.56·24-s − 0.689·25-s + 6.09·26-s − 5.55·27-s + 4.64·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.902·3-s + 0.5·4-s − 0.928·5-s + 0.638·6-s + 1.75·7-s + 0.353·8-s − 0.185·9-s − 0.656·10-s − 0.301·11-s + 0.451·12-s + 1.69·13-s + 1.24·14-s − 0.838·15-s + 0.250·16-s + 0.433·17-s − 0.130·18-s + 1.58·19-s − 0.464·20-s + 1.58·21-s − 0.213·22-s − 1.14·23-s + 0.319·24-s − 0.137·25-s + 1.19·26-s − 1.06·27-s + 0.878·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4334\)    =    \(2 \cdot 11 \cdot 197\)
Sign: $1$
Analytic conductor: \(34.6071\)
Root analytic conductor: \(5.88278\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4334,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.700045578\)
\(L(\frac12)\) \(\approx\) \(4.700045578\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
11 \( 1 + T \)
197 \( 1 + T \)
good3 \( 1 - 1.56T + 3T^{2} \)
5 \( 1 + 2.07T + 5T^{2} \)
7 \( 1 - 4.64T + 7T^{2} \)
13 \( 1 - 6.09T + 13T^{2} \)
17 \( 1 - 1.78T + 17T^{2} \)
19 \( 1 - 6.90T + 19T^{2} \)
23 \( 1 + 5.48T + 23T^{2} \)
29 \( 1 + 4.90T + 29T^{2} \)
31 \( 1 - 1.11T + 31T^{2} \)
37 \( 1 - 0.379T + 37T^{2} \)
41 \( 1 - 2.79T + 41T^{2} \)
43 \( 1 - 4.01T + 43T^{2} \)
47 \( 1 - 4.54T + 47T^{2} \)
53 \( 1 - 7.21T + 53T^{2} \)
59 \( 1 - 7.85T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 3.73T + 67T^{2} \)
71 \( 1 - 2.38T + 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 6.38T + 83T^{2} \)
89 \( 1 + 1.41T + 89T^{2} \)
97 \( 1 + 9.85T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.173343861602155990695009638728, −7.77940226925979763368861147110, −7.27209613352403066750754809536, −5.78685840562673075971684811731, −5.52172425192466719296775365276, −4.34011685815381032358762375596, −3.85717285928980855599740515046, −3.14183002297278505540552761916, −2.08467510717804119193039649781, −1.16129431285513574240643744359, 1.16129431285513574240643744359, 2.08467510717804119193039649781, 3.14183002297278505540552761916, 3.85717285928980855599740515046, 4.34011685815381032358762375596, 5.52172425192466719296775365276, 5.78685840562673075971684811731, 7.27209613352403066750754809536, 7.77940226925979763368861147110, 8.173343861602155990695009638728

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