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Properties

Label 2-1305-5.4-c1-0-43
Degree $2$
Conductor $1305$
Sign $-0.343 + 0.939i$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
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  − 1.78i·2-s − 1.19·4-s + (0.766 − 2.10i)5-s + 4.04i·7-s − 1.43i·8-s + (−3.75 − 1.37i)10-s + 4.80·11-s + 0.533i·13-s + 7.23·14-s − 4.96·16-s + 0.299i·17-s + 6.02·19-s + (−0.919 + 2.51i)20-s − 8.60i·22-s + 0.379i·23-s + ⋯
L(s)  = 1  − 1.26i·2-s − 0.599·4-s + (0.343 − 0.939i)5-s + 1.52i·7-s − 0.506i·8-s + (−1.18 − 0.433i)10-s + 1.45·11-s + 0.147i·13-s + 1.93·14-s − 1.24·16-s + 0.0727i·17-s + 1.38·19-s + (−0.205 + 0.562i)20-s − 1.83i·22-s + 0.0790i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $-0.343 + 0.939i$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ -0.343 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.064913625\)
\(L(\frac12)\) \(\approx\) \(2.064913625\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.766 + 2.10i)T \)
29 \( 1 - T \)
good2 \( 1 + 1.78iT - 2T^{2} \)
7 \( 1 - 4.04iT - 7T^{2} \)
11 \( 1 - 4.80T + 11T^{2} \)
13 \( 1 - 0.533iT - 13T^{2} \)
17 \( 1 - 0.299iT - 17T^{2} \)
19 \( 1 - 6.02T + 19T^{2} \)
23 \( 1 - 0.379iT - 23T^{2} \)
31 \( 1 - 9.14T + 31T^{2} \)
37 \( 1 + 8.51iT - 37T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 - 6.01iT - 43T^{2} \)
47 \( 1 - 0.299iT - 47T^{2} \)
53 \( 1 + 11.8iT - 53T^{2} \)
59 \( 1 + 6.53T + 59T^{2} \)
61 \( 1 + 0.755T + 61T^{2} \)
67 \( 1 + 3.95iT - 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 11.0iT - 73T^{2} \)
79 \( 1 + 7.01T + 79T^{2} \)
83 \( 1 - 6.15iT - 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + 2.41iT - 97T^{2} \)
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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

−9.472470936003723018789283884885, −8.987022391669807709017955490149, −8.178665964849230998827765351108, −6.75773805754557290043425163874, −5.93232686943844731712152901168, −5.01872532869100627633828308371, −4.03797670823268630167315148707, −2.97258678067441179227819774724, −1.96904144178307209467216579214, −1.07047449492684355151368099466, 1.27618046929398321741596015633, 2.97644345060059365473652550203, 4.00296084311983286710328378482, 4.97260137234829959162209482821, 6.19372685747963115065652775221, 6.62351818834925643527670756855, 7.31118594307119829820958340195, 7.85520692054445231688601299721, 8.971507349167364162100591778086, 9.875292024189542929700477882329

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