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Properties

Label 2-1305-145.144-c1-0-60
Degree $2$
Conductor $1305$
Sign $0.0830 + 0.996i$
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  + 2·2-s + 2·4-s + (−2 + i)5-s − 4i·7-s + (−4 + 2i)10-s + i·11-s − 2i·13-s − 8i·14-s − 4·16-s + 6·17-s − 4i·19-s + (−4 + 2i)20-s + 2i·22-s − 9i·23-s + (3 − 4i)25-s − 4i·26-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + (−0.894 + 0.447i)5-s − 1.51i·7-s + (−1.26 + 0.632i)10-s + 0.301i·11-s − 0.554i·13-s − 2.13i·14-s − 16-s + 1.45·17-s − 0.917i·19-s + (−0.894 + 0.447i)20-s + 0.426i·22-s − 1.87i·23-s + (0.600 − 0.800i)25-s − 0.784i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0830 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0830 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.0830 + 0.996i$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ 0.0830 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.453481913\)
\(L(\frac12)\) \(\approx\) \(2.453481913\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - i)T \)
29 \( 1 + (-2 + 5i)T \)
good2 \( 1 - 2T + 2T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 9iT - 23T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 9iT - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 6iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 15T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 7iT - 83T^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 - 11T + 97T^{2} \)
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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

−9.738429152327209098155837035283, −8.302843257897433570293909223490, −7.60973984109287156553467338298, −6.83579944325227858511132406863, −6.14885502234064744496038314249, −4.80012059859727431091523076309, −4.37797694144427953366951522619, −3.47663691456538662284798495539, −2.76150253355720034799778383644, −0.64223607711012492924852643119, 1.77088019764797899931344369182, 3.28478753548417363019912845808, 3.59254007917073859749384693798, 4.95488288955330625006751518500, 5.40613569696574780957360186736, 6.13340633299569553057515089107, 7.27300529849285495151592315098, 8.207370617276444456773597116117, 8.950893201578666372078248978392, 9.705241165562487214321846287697

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