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Properties

Label 2-2394-133.132-c1-0-39
Degree $2$
Conductor $2394$
Sign $0.999 + 0.0394i$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
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 − 4-s − 1.63i·5-s + (1.91 − 1.82i)7-s − i·8-s + 1.63·10-s + 2.23·11-s − 0.801·13-s + (1.82 + 1.91i)14-s + 16-s + 2.71i·17-s + (−3.13 + 3.02i)19-s + 1.63i·20-s + 2.23i·22-s + 6.49·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.733i·5-s + (0.722 − 0.691i)7-s − 0.353i·8-s + 0.518·10-s + 0.672·11-s − 0.222·13-s + (0.488 + 0.511i)14-s + 0.250·16-s + 0.658i·17-s + (−0.719 + 0.694i)19-s + 0.366i·20-s + 0.475i·22-s + 1.35·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $0.999 + 0.0394i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (1063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ 0.999 + 0.0394i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.897598319\)
\(L(\frac12)\) \(\approx\) \(1.897598319\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (-1.91 + 1.82i)T \)
19 \( 1 + (3.13 - 3.02i)T \)
good5 \( 1 + 1.63iT - 5T^{2} \)
11 \( 1 - 2.23T + 11T^{2} \)
13 \( 1 + 0.801T + 13T^{2} \)
17 \( 1 - 2.71iT - 17T^{2} \)
23 \( 1 - 6.49T + 23T^{2} \)
29 \( 1 - 3.82iT - 29T^{2} \)
31 \( 1 + 5.19T + 31T^{2} \)
37 \( 1 + 7.19iT - 37T^{2} \)
41 \( 1 - 6.99T + 41T^{2} \)
43 \( 1 - 7.13T + 43T^{2} \)
47 \( 1 + 1.91iT - 47T^{2} \)
53 \( 1 + 1.19iT - 53T^{2} \)
59 \( 1 - 4.59T + 59T^{2} \)
61 \( 1 + 0.934iT - 61T^{2} \)
67 \( 1 + 15.2iT - 67T^{2} \)
71 \( 1 + 1.19iT - 71T^{2} \)
73 \( 1 + 6.99iT - 73T^{2} \)
79 \( 1 + 0.503iT - 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 + 17.9T + 97T^{2} \)
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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

−8.985587869952850730017118600645, −8.126074998002885678726030543724, −7.45486850604653972121033735192, −6.74009341134090365207254570315, −5.81203561106869896787161397926, −5.00046492297733627119309885139, −4.31687883362458492482446319046, −3.55586337326353285855249073513, −1.85027646320313048413921972441, −0.809954292861860703411789714090, 1.07847807522177772978079447431, 2.38427770079802306333555473549, 2.89387507619629006767589991812, 4.11056355301484644798146987488, 4.87220105215265528832329376746, 5.73060120966821768241514275472, 6.74107217695598942205697450811, 7.39411222931616482645881127773, 8.442257463569396769382139968462, 9.055797988678626270122414671875

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