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Properties

Label 2-2394-21.20-c1-0-31
Degree $2$
Conductor $2394$
Sign $-0.0366 + 0.999i$
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 − 0.105·5-s + (2.10 + 1.60i)7-s + i·8-s + 0.105i·10-s − 5.08i·11-s − 2.42i·13-s + (1.60 − 2.10i)14-s + 16-s + 7.88·17-s − i·19-s + 0.105·20-s − 5.08·22-s − 1.53i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.0473·5-s + (0.794 + 0.606i)7-s + 0.353i·8-s + 0.0334i·10-s − 1.53i·11-s − 0.673i·13-s + (0.429 − 0.561i)14-s + 0.250·16-s + 1.91·17-s − 0.229i·19-s + 0.0236·20-s − 1.08·22-s − 0.319i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0366 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0366 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-0.0366 + 0.999i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (2015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ -0.0366 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.846882949\)
\(L(\frac12)\) \(\approx\) \(1.846882949\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 - 1.60i)T \)
19 \( 1 + iT \)
good5 \( 1 + 0.105T + 5T^{2} \)
11 \( 1 + 5.08iT - 11T^{2} \)
13 \( 1 + 2.42iT - 13T^{2} \)
17 \( 1 - 7.88T + 17T^{2} \)
23 \( 1 + 1.53iT - 23T^{2} \)
29 \( 1 - 5.47iT - 29T^{2} \)
31 \( 1 - 8.62iT - 31T^{2} \)
37 \( 1 + 1.01T + 37T^{2} \)
41 \( 1 - 1.74T + 41T^{2} \)
43 \( 1 - 0.616T + 43T^{2} \)
47 \( 1 - 1.27T + 47T^{2} \)
53 \( 1 + 10.6iT - 53T^{2} \)
59 \( 1 - 1.88T + 59T^{2} \)
61 \( 1 + 13.3iT - 61T^{2} \)
67 \( 1 - 7.69T + 67T^{2} \)
71 \( 1 + 15.4iT - 71T^{2} \)
73 \( 1 + 7.15iT - 73T^{2} \)
79 \( 1 + 6.21T + 79T^{2} \)
83 \( 1 + 1.69T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 6.42iT - 97T^{2} \)
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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

−8.659661233394619354730864113865, −8.238588185661298203106510661861, −7.52241185840476458402546355735, −6.18475171286534306623326906527, −5.44245214665265146299491267137, −4.94263235150833834401856724760, −3.47271268558778824245325928198, −3.15263743974935680149906390517, −1.81258296542435790262294216862, −0.74333253651756691049403396148, 1.17189863665666641475277661355, 2.28048041324118668604310592791, 3.92412020606054356645456794591, 4.29911875919739257084694614263, 5.31904349475351754876956014858, 5.99382564297384089996417753863, 7.11625639445208901349920963098, 7.62771910926456477081247885658, 8.014110539952442782928206407294, 9.181688613308068713034989441343

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