Properties

Label 2-1336-1.1-c1-0-7
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  + 0.571·3-s − 2.42·5-s − 3.89·7-s − 2.67·9-s + 1.06·11-s + 5.87·13-s − 1.38·15-s + 4.69·17-s − 4.56·19-s − 2.22·21-s + 6.88·23-s + 0.867·25-s − 3.24·27-s − 4.73·29-s + 5.09·31-s + 0.606·33-s + 9.43·35-s + 7.53·37-s + 3.35·39-s − 4.59·41-s + 7.94·43-s + 6.47·45-s + 10.0·47-s + 8.17·49-s + 2.68·51-s + 7.90·53-s − 2.57·55-s + ⋯
L(s)  = 1  + 0.330·3-s − 1.08·5-s − 1.47·7-s − 0.890·9-s + 0.319·11-s + 1.62·13-s − 0.357·15-s + 1.13·17-s − 1.04·19-s − 0.486·21-s + 1.43·23-s + 0.173·25-s − 0.624·27-s − 0.878·29-s + 0.914·31-s + 0.105·33-s + 1.59·35-s + 1.23·37-s + 0.537·39-s − 0.718·41-s + 1.21·43-s + 0.965·45-s + 1.46·47-s + 1.16·49-s + 0.375·51-s + 1.08·53-s − 0.346·55-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

\(L(1)\) \(\approx\) \(1.170756110\)
\(L(\frac12)\) \(\approx\) \(1.170756110\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.571T + 3T^{2} \)
5 \( 1 + 2.42T + 5T^{2} \)
7 \( 1 + 3.89T + 7T^{2} \)
11 \( 1 - 1.06T + 11T^{2} \)
13 \( 1 - 5.87T + 13T^{2} \)
17 \( 1 - 4.69T + 17T^{2} \)
19 \( 1 + 4.56T + 19T^{2} \)
23 \( 1 - 6.88T + 23T^{2} \)
29 \( 1 + 4.73T + 29T^{2} \)
31 \( 1 - 5.09T + 31T^{2} \)
37 \( 1 - 7.53T + 37T^{2} \)
41 \( 1 + 4.59T + 41T^{2} \)
43 \( 1 - 7.94T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 7.90T + 53T^{2} \)
59 \( 1 + 7.93T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 + 6.91T + 67T^{2} \)
71 \( 1 + 5.67T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 + 5.05T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−9.400725425692048738233082706666, −8.797462330580167734176756945206, −8.124450220077516943744114970910, −7.21243266690833231437888507554, −6.28413122699739751769307866081, −5.68257123124367428124744754022, −4.09010009179829999778041986982, −3.53230383712891784405417157931, −2.76652621113286438428306661756, −0.76679874117256961077544409395, 0.76679874117256961077544409395, 2.76652621113286438428306661756, 3.53230383712891784405417157931, 4.09010009179829999778041986982, 5.68257123124367428124744754022, 6.28413122699739751769307866081, 7.21243266690833231437888507554, 8.124450220077516943744114970910, 8.797462330580167734176756945206, 9.400725425692048738233082706666

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