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Properties

Label 2-2007-223.103-c0-0-0
Degree $2$
Conductor $2007$
Sign $0.402 - 0.915i$
Analytic cond. $1.00162$
Root an. cond. $1.00081$
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  + (−0.372 + 0.927i)4-s + (1.42 − 0.244i)7-s + (−0.953 + 1.70i)13-s + (−0.721 − 0.691i)16-s + (1.64 + 0.581i)19-s + (−0.911 − 0.411i)25-s + (−0.304 + 1.41i)28-s + (0.925 − 1.25i)31-s + (0.535 + 1.06i)37-s + (−1.71 + 0.776i)43-s + (1.02 − 0.361i)49-s + (−1.22 − 1.52i)52-s + (−0.164 + 0.295i)61-s + (0.911 − 0.411i)64-s + (1.84 + 0.739i)67-s + ⋯
L(s)  = 1  + (−0.372 + 0.927i)4-s + (1.42 − 0.244i)7-s + (−0.953 + 1.70i)13-s + (−0.721 − 0.691i)16-s + (1.64 + 0.581i)19-s + (−0.911 − 0.411i)25-s + (−0.304 + 1.41i)28-s + (0.925 − 1.25i)31-s + (0.535 + 1.06i)37-s + (−1.71 + 0.776i)43-s + (1.02 − 0.361i)49-s + (−1.22 − 1.52i)52-s + (−0.164 + 0.295i)61-s + (0.911 − 0.411i)64-s + (1.84 + 0.739i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $0.402 - 0.915i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ 0.402 - 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.185973655\)
\(L(\frac12)\) \(\approx\) \(1.185973655\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( 1 + (-0.911 - 0.411i)T \)
good2 \( 1 + (0.372 - 0.927i)T^{2} \)
5 \( 1 + (0.911 + 0.411i)T^{2} \)
7 \( 1 + (-1.42 + 0.244i)T + (0.942 - 0.333i)T^{2} \)
11 \( 1 + (0.594 + 0.803i)T^{2} \)
13 \( 1 + (0.953 - 1.70i)T + (-0.524 - 0.851i)T^{2} \)
17 \( 1 + (-0.594 - 0.803i)T^{2} \)
19 \( 1 + (-1.64 - 0.581i)T + (0.778 + 0.628i)T^{2} \)
23 \( 1 + (0.911 + 0.411i)T^{2} \)
29 \( 1 + (0.0424 + 0.999i)T^{2} \)
31 \( 1 + (-0.925 + 1.25i)T + (-0.292 - 0.956i)T^{2} \)
37 \( 1 + (-0.535 - 1.06i)T + (-0.594 + 0.803i)T^{2} \)
41 \( 1 + (0.873 - 0.487i)T^{2} \)
43 \( 1 + (1.71 - 0.776i)T + (0.660 - 0.750i)T^{2} \)
47 \( 1 + (0.778 - 0.628i)T^{2} \)
53 \( 1 + (0.985 + 0.169i)T^{2} \)
59 \( 1 + (0.967 + 0.251i)T^{2} \)
61 \( 1 + (0.164 - 0.295i)T + (-0.524 - 0.851i)T^{2} \)
67 \( 1 + (-1.84 - 0.739i)T + (0.721 + 0.691i)T^{2} \)
71 \( 1 + (0.967 + 0.251i)T^{2} \)
73 \( 1 + (0.580 - 0.468i)T + (0.210 - 0.977i)T^{2} \)
79 \( 1 + (0.207 + 0.797i)T + (-0.873 + 0.487i)T^{2} \)
83 \( 1 + (0.660 + 0.750i)T^{2} \)
89 \( 1 + (-0.911 + 0.411i)T^{2} \)
97 \( 1 + (-0.492 + 0.106i)T + (0.911 - 0.411i)T^{2} \)
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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

−9.575555532973516822537672736950, −8.519309825086378910931689858386, −7.901416035752528696196140213987, −7.41928480859222229231391366703, −6.50254364015130013423813573930, −5.18997951540404126994818086049, −4.55465640842408932823643416836, −3.91813163862729998229432038465, −2.65483659552169808562031876267, −1.58578298358945146030171914642, 0.957929199288751585038065112332, 2.10728805280077071794185928944, 3.28691913832826085064133278562, 4.67948185834786377012516685826, 5.25170872108007234830340321065, 5.60721108582803073329364671218, 6.92984292559399976387873717432, 7.79013589124664338979708455843, 8.327700033978624348240420441560, 9.308656200740122253704001499838

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