Properties

Label 2-1334-1.1-c1-0-27
Degree $2$
Conductor $1334$
Sign $1$
Analytic cond. $10.6520$
Root an. cond. $3.26374$
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 + 2.61·3-s + 4-s + 0.316·5-s + 2.61·6-s − 4.23·7-s + 8-s + 3.85·9-s + 0.316·10-s + 2.97·11-s + 2.61·12-s + 3.99·13-s − 4.23·14-s + 0.827·15-s + 16-s + 4.57·17-s + 3.85·18-s + 1.78·19-s + 0.316·20-s − 11.0·21-s + 2.97·22-s + 23-s + 2.61·24-s − 4.90·25-s + 3.99·26-s + 2.22·27-s − 4.23·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.51·3-s + 0.5·4-s + 0.141·5-s + 1.06·6-s − 1.60·7-s + 0.353·8-s + 1.28·9-s + 0.0999·10-s + 0.897·11-s + 0.755·12-s + 1.10·13-s − 1.13·14-s + 0.213·15-s + 0.250·16-s + 1.10·17-s + 0.907·18-s + 0.409·19-s + 0.0706·20-s − 2.42·21-s + 0.634·22-s + 0.208·23-s + 0.534·24-s − 0.980·25-s + 0.784·26-s + 0.428·27-s − 0.800·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1334\)    =    \(2 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(10.6520\)
Root analytic conductor: \(3.26374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1334,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.147901594\)
\(L(\frac12)\) \(\approx\) \(4.147901594\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
good3 \( 1 - 2.61T + 3T^{2} \)
5 \( 1 - 0.316T + 5T^{2} \)
7 \( 1 + 4.23T + 7T^{2} \)
11 \( 1 - 2.97T + 11T^{2} \)
13 \( 1 - 3.99T + 13T^{2} \)
17 \( 1 - 4.57T + 17T^{2} \)
19 \( 1 - 1.78T + 19T^{2} \)
31 \( 1 + 0.0125T + 31T^{2} \)
37 \( 1 + 8.58T + 37T^{2} \)
41 \( 1 - 5.02T + 41T^{2} \)
43 \( 1 + 5.02T + 43T^{2} \)
47 \( 1 + 4.22T + 47T^{2} \)
53 \( 1 + 6.96T + 53T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 + 8.71T + 61T^{2} \)
67 \( 1 - 9.45T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 - 3.99T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + 8.37T + 89T^{2} \)
97 \( 1 - 5.75T + 97T^{2} \)
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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

−9.649916319734872169090021732015, −8.849339782772852946745406891399, −8.062415485100264427760307326627, −7.07013449371422524308704224904, −6.38895093703613101648192053763, −5.53100700954809651117493626303, −3.91840633406307895763170959615, −3.52806972895713462626475977991, −2.83670437235886265413095341948, −1.50550447042976015404119915262, 1.50550447042976015404119915262, 2.83670437235886265413095341948, 3.52806972895713462626475977991, 3.91840633406307895763170959615, 5.53100700954809651117493626303, 6.38895093703613101648192053763, 7.07013449371422524308704224904, 8.062415485100264427760307326627, 8.849339782772852946745406891399, 9.649916319734872169090021732015

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