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Properties

Label 2-1305-5.4-c1-0-20
Degree $2$
Conductor $1305$
Sign $-0.870 - 0.492i$
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  + 1.35i·2-s + 0.163·4-s + (1.94 + 1.10i)5-s + 1.26i·7-s + 2.93i·8-s + (−1.49 + 2.63i)10-s − 0.474·11-s − 0.407i·13-s − 1.72·14-s − 3.64·16-s + 2.97i·17-s − 5.80·19-s + (0.318 + 0.180i)20-s − 0.643i·22-s + 3.06i·23-s + ⋯
L(s)  = 1  + 0.958i·2-s + 0.0818·4-s + (0.870 + 0.492i)5-s + 0.479i·7-s + 1.03i·8-s + (−0.472 + 0.833i)10-s − 0.143·11-s − 0.113i·13-s − 0.459·14-s − 0.911·16-s + 0.720i·17-s − 1.33·19-s + (0.0711 + 0.0403i)20-s − 0.137i·22-s + 0.639i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $-0.870 - 0.492i$
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: \((2,\ 1305,\ (\ :1/2),\ -0.870 - 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.980309993\)
\(L(\frac12)\) \(\approx\) \(1.980309993\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 - 1.10i)T \)
29 \( 1 - T \)
good2 \( 1 - 1.35iT - 2T^{2} \)
7 \( 1 - 1.26iT - 7T^{2} \)
11 \( 1 + 0.474T + 11T^{2} \)
13 \( 1 + 0.407iT - 13T^{2} \)
17 \( 1 - 2.97iT - 17T^{2} \)
19 \( 1 + 5.80T + 19T^{2} \)
23 \( 1 - 3.06iT - 23T^{2} \)
31 \( 1 + 4.49T + 31T^{2} \)
37 \( 1 + 7.20iT - 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 + 1.03iT - 43T^{2} \)
47 \( 1 - 2.97iT - 47T^{2} \)
53 \( 1 + 7.87iT - 53T^{2} \)
59 \( 1 - 4.45T + 59T^{2} \)
61 \( 1 + 1.71T + 61T^{2} \)
67 \( 1 - 15.7iT - 67T^{2} \)
71 \( 1 - 9.00T + 71T^{2} \)
73 \( 1 + 0.677iT - 73T^{2} \)
79 \( 1 + 1.63T + 79T^{2} \)
83 \( 1 - 15.5iT - 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + 7.10iT - 97T^{2} \)
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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

−9.889539474138797863196703279674, −8.983486295358946821500639253735, −8.308577684367138954328988559156, −7.35353360105108810321345911410, −6.67083206864558965782107556901, −5.79701627798307165513313971812, −5.50671901705582352262134713737, −4.09815188480945160639777787135, −2.65862703344932060887531268490, −1.90887401627506906680505624312, 0.78467214796261848332306693812, 1.98101873579884030117542131683, 2.77208572178030060856977898751, 4.03686849184595596934415526877, 4.83671411702892654159643289199, 6.05576814771792792722015965583, 6.71651921581079084622814345635, 7.69879910571331451791443042526, 8.840856613965620671534627760209, 9.449818913700568591797503197509

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