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Properties

Label 2-1334-29.28-c1-0-55
Degree $2$
Conductor $1334$
Sign $0.505 - 0.862i$
Analytic cond. $10.6520$
Root an. cond. $3.26374$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 3.33i·3-s − 4-s + 3.07·5-s − 3.33·6-s − 2.74·7-s + i·8-s − 8.09·9-s − 3.07i·10-s + 5.14i·11-s + 3.33i·12-s − 5.53·13-s + 2.74i·14-s − 10.2i·15-s + 16-s + 0.785i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.92i·3-s − 0.5·4-s + 1.37·5-s − 1.35·6-s − 1.03·7-s + 0.353i·8-s − 2.69·9-s − 0.972i·10-s + 1.55i·11-s + 0.961i·12-s − 1.53·13-s + 0.732i·14-s − 2.64i·15-s + 0.250·16-s + 0.190i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1334\)    =    \(2 \cdot 23 \cdot 29\)
Sign: $0.505 - 0.862i$
Analytic conductor: \(10.6520\)
Root analytic conductor: \(3.26374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1334} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1334,\ (\ :1/2),\ 0.505 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1713040488\)
\(L(\frac12)\) \(\approx\) \(0.1713040488\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
23 \( 1 + T \)
29 \( 1 + (4.64 + 2.72i)T \)
good3 \( 1 + 3.33iT - 3T^{2} \)
5 \( 1 - 3.07T + 5T^{2} \)
7 \( 1 + 2.74T + 7T^{2} \)
11 \( 1 - 5.14iT - 11T^{2} \)
13 \( 1 + 5.53T + 13T^{2} \)
17 \( 1 - 0.785iT - 17T^{2} \)
19 \( 1 - 0.159iT - 19T^{2} \)
31 \( 1 + 1.07iT - 31T^{2} \)
37 \( 1 - 4.40iT - 37T^{2} \)
41 \( 1 + 9.96iT - 41T^{2} \)
43 \( 1 - 0.306iT - 43T^{2} \)
47 \( 1 - 5.50iT - 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 13.9iT - 61T^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 + 4.87T + 71T^{2} \)
73 \( 1 + 4.46iT - 73T^{2} \)
79 \( 1 + 9.52iT - 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 - 3.66iT - 89T^{2} \)
97 \( 1 + 10.7iT - 97T^{2} \)
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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

−9.354718521958171157179202828781, −7.945891193044814822205319953283, −7.23445441007066737378907647230, −6.52778847107246552813560027466, −5.81788454383649683108968788557, −4.85031491718019074421081173825, −3.05385100356501608380194496447, −2.15114603413714935119549412147, −1.77670285178451971474940631402, −0.06284137568560965753683077407, 2.72128962843253859181961091938, 3.40977163009202688528565839248, 4.56446178330066011641328266299, 5.48207026951829530693972710284, 5.81191739483315588844791254679, 6.70879182523895141383097914086, 8.131329178471987894525645972691, 9.106477300238952742989547294167, 9.473709884868273263951930323243, 9.970134402721094439473500543628

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