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Properties

Label 2-2007-223.207-c0-0-0
Degree $2$
Conductor $2007$
Sign $-0.962 - 0.270i$
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  + (−0.210 + 0.977i)4-s + (−1.76 − 0.459i)7-s + (1.23 + 1.28i)13-s + (−0.911 − 0.411i)16-s + (−1.26 + 0.704i)19-s + (−0.594 − 0.803i)25-s + (0.820 − 1.62i)28-s + (−1.03 + 0.177i)31-s + (−1.96 + 0.167i)37-s + (−1.03 + 1.40i)43-s + (2.02 + 1.13i)49-s + (−1.52 + 0.936i)52-s + (−0.348 − 0.363i)61-s + (0.594 − 0.803i)64-s + (−1.09 − 0.235i)67-s + ⋯
L(s)  = 1  + (−0.210 + 0.977i)4-s + (−1.76 − 0.459i)7-s + (1.23 + 1.28i)13-s + (−0.911 − 0.411i)16-s + (−1.26 + 0.704i)19-s + (−0.594 − 0.803i)25-s + (0.820 − 1.62i)28-s + (−1.03 + 0.177i)31-s + (−1.96 + 0.167i)37-s + (−1.03 + 1.40i)43-s + (2.02 + 1.13i)49-s + (−1.52 + 0.936i)52-s + (−0.348 − 0.363i)61-s + (0.594 − 0.803i)64-s + (−1.09 − 0.235i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $-0.962 - 0.270i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ -0.962 - 0.270i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3855001424\)
\(L(\frac12)\) \(\approx\) \(0.3855001424\)
\(L(1)\) 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 \( 1 + (-0.594 - 0.803i)T \)
good2 \( 1 + (0.210 - 0.977i)T^{2} \)
5 \( 1 + (0.594 + 0.803i)T^{2} \)
7 \( 1 + (1.76 + 0.459i)T + (0.873 + 0.487i)T^{2} \)
11 \( 1 + (-0.985 - 0.169i)T^{2} \)
13 \( 1 + (-1.23 - 1.28i)T + (-0.0424 + 0.999i)T^{2} \)
17 \( 1 + (0.985 + 0.169i)T^{2} \)
19 \( 1 + (1.26 - 0.704i)T + (0.524 - 0.851i)T^{2} \)
23 \( 1 + (0.594 + 0.803i)T^{2} \)
29 \( 1 + (0.660 + 0.750i)T^{2} \)
31 \( 1 + (1.03 - 0.177i)T + (0.942 - 0.333i)T^{2} \)
37 \( 1 + (1.96 - 0.167i)T + (0.985 - 0.169i)T^{2} \)
41 \( 1 + (-0.721 - 0.691i)T^{2} \)
43 \( 1 + (1.03 - 1.40i)T + (-0.292 - 0.956i)T^{2} \)
47 \( 1 + (0.524 + 0.851i)T^{2} \)
53 \( 1 + (-0.967 + 0.251i)T^{2} \)
59 \( 1 + (-0.372 - 0.927i)T^{2} \)
61 \( 1 + (0.348 + 0.363i)T + (-0.0424 + 0.999i)T^{2} \)
67 \( 1 + (1.09 + 0.235i)T + (0.911 + 0.411i)T^{2} \)
71 \( 1 + (-0.372 - 0.927i)T^{2} \)
73 \( 1 + (0.220 + 0.358i)T + (-0.450 + 0.892i)T^{2} \)
79 \( 1 + (-1.49 - 0.599i)T + (0.721 + 0.691i)T^{2} \)
83 \( 1 + (-0.292 + 0.956i)T^{2} \)
89 \( 1 + (-0.594 + 0.803i)T^{2} \)
97 \( 1 + (-1.65 + 0.835i)T + (0.594 - 0.803i)T^{2} \)
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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

−9.556211212733270291764533191059, −8.888547179762505579202089734420, −8.296933982528677274035581531102, −7.23587262011294972129453809691, −6.53681733255188080407849027473, −6.11109794517412120842949775998, −4.54263206154416634179158519121, −3.72852172934556428129146228795, −3.34546060646131210870288573850, −1.96400355418254318396037576005, 0.26043027754146785053525788074, 1.90929697390197316661171237413, 3.17508180243342868902898714855, 3.85913650909193011072162196098, 5.25897721580972413948267261053, 5.82774062092763788904813961186, 6.45439401335563696998607148144, 7.18757352940840768499895531825, 8.600927106126534939335958782618, 8.958580070015930264125662773944

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