Properties

Label 2-4334-1.1-c1-0-121
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.67·3-s + 4-s + 2.35·5-s + 1.67·6-s + 2.46·7-s + 8-s − 0.186·9-s + 2.35·10-s + 11-s + 1.67·12-s + 2.53·13-s + 2.46·14-s + 3.95·15-s + 16-s + 6.04·17-s − 0.186·18-s − 7.69·19-s + 2.35·20-s + 4.13·21-s + 22-s + 1.49·23-s + 1.67·24-s + 0.555·25-s + 2.53·26-s − 5.34·27-s + 2.46·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.968·3-s + 0.5·4-s + 1.05·5-s + 0.684·6-s + 0.932·7-s + 0.353·8-s − 0.0621·9-s + 0.745·10-s + 0.301·11-s + 0.484·12-s + 0.703·13-s + 0.659·14-s + 1.02·15-s + 0.250·16-s + 1.46·17-s − 0.0439·18-s − 1.76·19-s + 0.527·20-s + 0.903·21-s + 0.213·22-s + 0.312·23-s + 0.342·24-s + 0.111·25-s + 0.497·26-s − 1.02·27-s + 0.466·28-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

\(L(1)\) \(\approx\) \(6.082922400\)
\(L(\frac12)\) \(\approx\) \(6.082922400\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.67T + 3T^{2} \)
5 \( 1 - 2.35T + 5T^{2} \)
7 \( 1 - 2.46T + 7T^{2} \)
13 \( 1 - 2.53T + 13T^{2} \)
17 \( 1 - 6.04T + 17T^{2} \)
19 \( 1 + 7.69T + 19T^{2} \)
23 \( 1 - 1.49T + 23T^{2} \)
29 \( 1 - 4.00T + 29T^{2} \)
31 \( 1 - 4.57T + 31T^{2} \)
37 \( 1 + 7.78T + 37T^{2} \)
41 \( 1 + 0.621T + 41T^{2} \)
43 \( 1 + 9.10T + 43T^{2} \)
47 \( 1 - 6.95T + 47T^{2} \)
53 \( 1 + 0.332T + 53T^{2} \)
59 \( 1 + 3.52T + 59T^{2} \)
61 \( 1 + 5.69T + 61T^{2} \)
67 \( 1 - 4.81T + 67T^{2} \)
71 \( 1 - 0.267T + 71T^{2} \)
73 \( 1 + 4.99T + 73T^{2} \)
79 \( 1 + 6.75T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 0.987T + 89T^{2} \)
97 \( 1 + 5.03T + 97T^{2} \)
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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

−8.392180292013967735841237533999, −7.81988288970604686048083593731, −6.75830960356846319301969504864, −6.08130015017882855035221363934, −5.42651293154161979856349135375, −4.61428674405014367157656413450, −3.71489988045253011213418098366, −2.93819232624874331905906844135, −2.04572326140218207659458360790, −1.41598031516847548275495126796, 1.41598031516847548275495126796, 2.04572326140218207659458360790, 2.93819232624874331905906844135, 3.71489988045253011213418098366, 4.61428674405014367157656413450, 5.42651293154161979856349135375, 6.08130015017882855035221363934, 6.75830960356846319301969504864, 7.81988288970604686048083593731, 8.392180292013967735841237533999

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