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Properties

Label 2-1423-1423.1422-c0-0-0
Degree $2$
Conductor $1423$
Sign $-1$
Analytic cond. $0.710169$
Root an. cond. $0.842715$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.41i·3-s + 1.41i·5-s − 1.41i·6-s + 8-s − 1.00·9-s − 1.41i·10-s + 1.41i·11-s − 1.41i·13-s − 2.00·15-s − 16-s + 1.41i·17-s + 1.00·18-s + 1.41i·19-s − 1.41i·22-s + 23-s + ⋯
L(s)  = 1  − 2-s + 1.41i·3-s + 1.41i·5-s − 1.41i·6-s + 8-s − 1.00·9-s − 1.41i·10-s + 1.41i·11-s − 1.41i·13-s − 2.00·15-s − 16-s + 1.41i·17-s + 1.00·18-s + 1.41i·19-s − 1.41i·22-s + 23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1423 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1423 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1423\)
Sign: $-1$
Analytic conductor: \(0.710169\)
Root analytic conductor: \(0.842715\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1423} (1422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1423,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5620694925\)
\(L(\frac12)\) \(\approx\) \(0.5620694925\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1423 \( 1 + T \)
good2 \( 1 + T + T^{2} \)
3 \( 1 - 1.41iT - T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.24042953437096268702333603027, −9.561498547068361088177944068064, −8.766775235690545652301799067882, −7.75754794788679335794160914617, −7.27117644796417620035132447558, −6.03355680749288262362217567283, −5.06877051523367677476645966802, −4.02929177106859037531323858594, −3.38235673822488915399128392032, −1.99756817523052614822497733042, 0.74339450222544620683082332076, 1.35354136271074698491992754940, 2.72975671087499469799615958808, 4.49125002678235357906382596931, 5.11444630555201034434460688817, 6.40156500916534450295667440460, 7.11522617015590980165763529640, 7.87536767148250116207455514995, 8.780900599682199735548913228679, 8.919045086754456038749689281434

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