Properties

Label 2-1334-1.1-c1-0-29
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.35·3-s + 4-s + 1.35·5-s + 1.35·6-s − 0.960·7-s − 8-s − 1.16·9-s − 1.35·10-s + 2.31·11-s − 1.35·12-s + 0.313·13-s + 0.960·14-s − 1.83·15-s + 16-s − 2.84·17-s + 1.16·18-s + 0.816·19-s + 1.35·20-s + 1.29·21-s − 2.31·22-s − 23-s + 1.35·24-s − 3.16·25-s − 0.313·26-s + 5.64·27-s − 0.960·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.781·3-s + 0.5·4-s + 0.605·5-s + 0.552·6-s − 0.363·7-s − 0.353·8-s − 0.389·9-s − 0.427·10-s + 0.697·11-s − 0.390·12-s + 0.0869·13-s + 0.256·14-s − 0.472·15-s + 0.250·16-s − 0.691·17-s + 0.275·18-s + 0.187·19-s + 0.302·20-s + 0.283·21-s − 0.493·22-s − 0.208·23-s + 0.276·24-s − 0.633·25-s − 0.0615·26-s + 1.08·27-s − 0.181·28-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: \(1\)
Selberg data: \((2,\ 1334,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.35T + 3T^{2} \)
5 \( 1 - 1.35T + 5T^{2} \)
7 \( 1 + 0.960T + 7T^{2} \)
11 \( 1 - 2.31T + 11T^{2} \)
13 \( 1 - 0.313T + 13T^{2} \)
17 \( 1 + 2.84T + 17T^{2} \)
19 \( 1 - 0.816T + 19T^{2} \)
31 \( 1 + 5.68T + 31T^{2} \)
37 \( 1 - 6.29T + 37T^{2} \)
41 \( 1 + 7.58T + 41T^{2} \)
43 \( 1 - 3.94T + 43T^{2} \)
47 \( 1 + 1.24T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 4.62T + 59T^{2} \)
61 \( 1 + 6.96T + 61T^{2} \)
67 \( 1 - 0.849T + 67T^{2} \)
71 \( 1 + 4.33T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + 3.01T + 79T^{2} \)
83 \( 1 - 0.366T + 83T^{2} \)
89 \( 1 + 5.21T + 89T^{2} \)
97 \( 1 + 5.41T + 97T^{2} \)
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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

−9.249087586088838926819576859971, −8.636257253569641130430757731633, −7.54655838570794790981591910358, −6.60121496703643990231141691070, −6.06144937696711355189013464842, −5.29869615302320818252843018020, −4.01424333125866348756512151561, −2.73545943910343702383571569010, −1.51934110215646368009139690093, 0, 1.51934110215646368009139690093, 2.73545943910343702383571569010, 4.01424333125866348756512151561, 5.29869615302320818252843018020, 6.06144937696711355189013464842, 6.60121496703643990231141691070, 7.54655838570794790981591910358, 8.636257253569641130430757731633, 9.249087586088838926819576859971

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