Properties

Label 2-4334-1.1-c1-0-52
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 + 2.36·3-s + 4-s − 0.759·5-s − 2.36·6-s − 2.98·7-s − 8-s + 2.60·9-s + 0.759·10-s + 11-s + 2.36·12-s + 5.18·13-s + 2.98·14-s − 1.79·15-s + 16-s + 5.91·17-s − 2.60·18-s − 6.38·19-s − 0.759·20-s − 7.07·21-s − 22-s + 0.346·23-s − 2.36·24-s − 4.42·25-s − 5.18·26-s − 0.937·27-s − 2.98·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.36·3-s + 0.5·4-s − 0.339·5-s − 0.966·6-s − 1.13·7-s − 0.353·8-s + 0.867·9-s + 0.240·10-s + 0.301·11-s + 0.683·12-s + 1.43·13-s + 0.799·14-s − 0.464·15-s + 0.250·16-s + 1.43·17-s − 0.613·18-s − 1.46·19-s − 0.169·20-s − 1.54·21-s − 0.213·22-s + 0.0722·23-s − 0.483·24-s − 0.884·25-s − 1.01·26-s − 0.180·27-s − 0.565·28-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

\(L(1)\) \(\approx\) \(1.955956963\)
\(L(\frac12)\) \(\approx\) \(1.955956963\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.36T + 3T^{2} \)
5 \( 1 + 0.759T + 5T^{2} \)
7 \( 1 + 2.98T + 7T^{2} \)
13 \( 1 - 5.18T + 13T^{2} \)
17 \( 1 - 5.91T + 17T^{2} \)
19 \( 1 + 6.38T + 19T^{2} \)
23 \( 1 - 0.346T + 23T^{2} \)
29 \( 1 + 0.999T + 29T^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 - 9.58T + 37T^{2} \)
41 \( 1 - 2.84T + 41T^{2} \)
43 \( 1 - 5.35T + 43T^{2} \)
47 \( 1 - 8.37T + 47T^{2} \)
53 \( 1 - 5.58T + 53T^{2} \)
59 \( 1 + 0.419T + 59T^{2} \)
61 \( 1 + 5.87T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + 7.25T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + 7.40T + 79T^{2} \)
83 \( 1 - 2.07T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 - 6.10T + 97T^{2} \)
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

−8.346796700859677773440182937697, −7.914606094803990572532731551842, −7.20338756156285825761206893904, −6.23530367851039696876397562334, −5.83130876419354299825700380509, −4.07211479101216596435588752767, −3.67673052895441980173946338370, −2.90802332582345551477336879899, −2.03713707770588902964022449448, −0.820434378478122163520623753388, 0.820434378478122163520623753388, 2.03713707770588902964022449448, 2.90802332582345551477336879899, 3.67673052895441980173946338370, 4.07211479101216596435588752767, 5.83130876419354299825700380509, 6.23530367851039696876397562334, 7.20338756156285825761206893904, 7.914606094803990572532731551842, 8.346796700859677773440182937697

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