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Properties

Label 2-1305-5.4-c1-0-0
Degree $2$
Conductor $1305$
Sign $0.662 + 0.748i$
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  + 2.62i·2-s − 4.89·4-s + (1.48 + 1.67i)5-s − 1.33i·7-s − 7.58i·8-s + (−4.39 + 3.88i)10-s − 4.65·11-s + 6.90i·13-s + 3.51·14-s + 10.1·16-s − 6.12i·17-s − 3.89·19-s + (−7.24 − 8.18i)20-s − 12.2i·22-s − 2.21i·23-s + ⋯
L(s)  = 1  + 1.85i·2-s − 2.44·4-s + (0.662 + 0.748i)5-s − 0.506i·7-s − 2.68i·8-s + (−1.39 + 1.23i)10-s − 1.40·11-s + 1.91i·13-s + 0.940·14-s + 2.53·16-s − 1.48i·17-s − 0.894·19-s + (−1.62 − 1.83i)20-s − 2.60i·22-s − 0.461i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.662 + 0.748i$
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: \((2,\ 1305,\ (\ :1/2),\ 0.662 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2482310263\)
\(L(\frac12)\) \(\approx\) \(0.2482310263\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 1.67i)T \)
29 \( 1 + T \)
good2 \( 1 - 2.62iT - 2T^{2} \)
7 \( 1 + 1.33iT - 7T^{2} \)
11 \( 1 + 4.65T + 11T^{2} \)
13 \( 1 - 6.90iT - 13T^{2} \)
17 \( 1 + 6.12iT - 17T^{2} \)
19 \( 1 + 3.89T + 19T^{2} \)
23 \( 1 + 2.21iT - 23T^{2} \)
31 \( 1 + 6.88T + 31T^{2} \)
37 \( 1 - 5.79iT - 37T^{2} \)
41 \( 1 + 7.05T + 41T^{2} \)
43 \( 1 + 11.9iT - 43T^{2} \)
47 \( 1 + 6.12iT - 47T^{2} \)
53 \( 1 + 1.31iT - 53T^{2} \)
59 \( 1 - 6.20T + 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 - 3.00iT - 67T^{2} \)
71 \( 1 - 6.04T + 71T^{2} \)
73 \( 1 - 4.47iT - 73T^{2} \)
79 \( 1 - 1.31T + 79T^{2} \)
83 \( 1 + 4.20iT - 83T^{2} \)
89 \( 1 - 0.232T + 89T^{2} \)
97 \( 1 - 2.99iT - 97T^{2} \)
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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

−10.06746373921167285390170517636, −9.254664184254643594907595570391, −8.631727853183409176619702826335, −7.53460072091568692665478578689, −6.98000360586313253857879952667, −6.54157202810782545101699654112, −5.45183886374401532590691602962, −4.84138651782698870402439276655, −3.79513442689632940155174305499, −2.25798200554752673514102601346, 0.099821713690000123534637619815, 1.55004825545321641840850171508, 2.46098236379939347301758520425, 3.31672975751760788004955223548, 4.46783896101280288880227082675, 5.43310733355626401223214087170, 5.81182213837020018641081722676, 7.947531461622397250861305245828, 8.351377242605352902785455808710, 9.229355344394198297395347151350

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