Properties

Label 2-1336-1.1-c1-0-18
Degree $2$
Conductor $1336$
Sign $1$
Analytic cond. $10.6680$
Root an. cond. $3.26619$
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  + 2.42·3-s − 0.247·5-s + 2.30·7-s + 2.88·9-s + 0.611·11-s − 3.57·13-s − 0.599·15-s + 2.36·17-s + 2.58·19-s + 5.60·21-s + 6.55·23-s − 4.93·25-s − 0.283·27-s + 7.68·29-s + 9.82·31-s + 1.48·33-s − 0.571·35-s + 6.31·37-s − 8.66·39-s − 9.78·41-s + 0.757·43-s − 0.713·45-s − 9.70·47-s − 1.66·49-s + 5.73·51-s − 3.18·53-s − 0.151·55-s + ⋯
L(s)  = 1  + 1.40·3-s − 0.110·5-s + 0.872·7-s + 0.961·9-s + 0.184·11-s − 0.990·13-s − 0.154·15-s + 0.573·17-s + 0.593·19-s + 1.22·21-s + 1.36·23-s − 0.987·25-s − 0.0544·27-s + 1.42·29-s + 1.76·31-s + 0.258·33-s − 0.0965·35-s + 1.03·37-s − 1.38·39-s − 1.52·41-s + 0.115·43-s − 0.106·45-s − 1.41·47-s − 0.238·49-s + 0.802·51-s − 0.438·53-s − 0.0203·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1336\)    =    \(2^{3} \cdot 167\)
Sign: $1$
Analytic conductor: \(10.6680\)
Root analytic conductor: \(3.26619\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1336,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.945641323\)
\(L(\frac12)\) \(\approx\) \(2.945641323\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 + T \)
good3 \( 1 - 2.42T + 3T^{2} \)
5 \( 1 + 0.247T + 5T^{2} \)
7 \( 1 - 2.30T + 7T^{2} \)
11 \( 1 - 0.611T + 11T^{2} \)
13 \( 1 + 3.57T + 13T^{2} \)
17 \( 1 - 2.36T + 17T^{2} \)
19 \( 1 - 2.58T + 19T^{2} \)
23 \( 1 - 6.55T + 23T^{2} \)
29 \( 1 - 7.68T + 29T^{2} \)
31 \( 1 - 9.82T + 31T^{2} \)
37 \( 1 - 6.31T + 37T^{2} \)
41 \( 1 + 9.78T + 41T^{2} \)
43 \( 1 - 0.757T + 43T^{2} \)
47 \( 1 + 9.70T + 47T^{2} \)
53 \( 1 + 3.18T + 53T^{2} \)
59 \( 1 - 3.12T + 59T^{2} \)
61 \( 1 + 5.52T + 61T^{2} \)
67 \( 1 - 3.92T + 67T^{2} \)
71 \( 1 - 4.50T + 71T^{2} \)
73 \( 1 - 3.57T + 73T^{2} \)
79 \( 1 + 3.75T + 79T^{2} \)
83 \( 1 + 8.82T + 83T^{2} \)
89 \( 1 + 1.73T + 89T^{2} \)
97 \( 1 + 18.8T + 97T^{2} \)
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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

−9.697981743276125933063504867684, −8.598554236839461917311355387862, −8.109477736983377665114836965075, −7.49522216210988886191216763196, −6.54544263657249522550520338007, −5.15066662463000132800866170850, −4.47831319399468958949478847577, −3.26464358829150767714201538611, −2.57173354912922029586624860215, −1.35016191060245711884461783201, 1.35016191060245711884461783201, 2.57173354912922029586624860215, 3.26464358829150767714201538611, 4.47831319399468958949478847577, 5.15066662463000132800866170850, 6.54544263657249522550520338007, 7.49522216210988886191216763196, 8.109477736983377665114836965075, 8.598554236839461917311355387862, 9.697981743276125933063504867684

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