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Properties

Label 2-2007-223.155-c0-0-0
Degree $2$
Conductor $2007$
Sign $0.247 - 0.968i$
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  + (0.594 + 0.803i)4-s + (−0.422 + 0.405i)7-s + (−0.253 − 0.223i)13-s + (−0.292 + 0.956i)16-s + (0.0560 + 1.32i)19-s + (0.942 + 0.333i)25-s + (−0.577 − 0.0989i)28-s + (1.73 + 0.971i)31-s + (−1.68 − 0.439i)37-s + (0.0800 − 0.0282i)43-s + (−0.0278 + 0.656i)49-s + (0.0286 − 0.336i)52-s + (−1.03 − 0.914i)61-s + (−0.942 + 0.333i)64-s + (1.57 − 1.16i)67-s + ⋯
L(s)  = 1  + (0.594 + 0.803i)4-s + (−0.422 + 0.405i)7-s + (−0.253 − 0.223i)13-s + (−0.292 + 0.956i)16-s + (0.0560 + 1.32i)19-s + (0.942 + 0.333i)25-s + (−0.577 − 0.0989i)28-s + (1.73 + 0.971i)31-s + (−1.68 − 0.439i)37-s + (0.0800 − 0.0282i)43-s + (−0.0278 + 0.656i)49-s + (0.0286 − 0.336i)52-s + (−1.03 − 0.914i)61-s + (−0.942 + 0.333i)64-s + (1.57 − 1.16i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $0.247 - 0.968i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (1270, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ 0.247 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217657895\)
\(L(\frac12)\) \(\approx\) \(1.217657895\)
\(L(1)\) 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 \( 1 + (0.942 + 0.333i)T \)
good2 \( 1 + (-0.594 - 0.803i)T^{2} \)
5 \( 1 + (-0.942 - 0.333i)T^{2} \)
7 \( 1 + (0.422 - 0.405i)T + (0.0424 - 0.999i)T^{2} \)
11 \( 1 + (-0.873 + 0.487i)T^{2} \)
13 \( 1 + (0.253 + 0.223i)T + (0.127 + 0.991i)T^{2} \)
17 \( 1 + (0.873 - 0.487i)T^{2} \)
19 \( 1 + (-0.0560 - 1.32i)T + (-0.996 + 0.0848i)T^{2} \)
23 \( 1 + (-0.942 - 0.333i)T^{2} \)
29 \( 1 + (-0.828 - 0.559i)T^{2} \)
31 \( 1 + (-1.73 - 0.971i)T + (0.524 + 0.851i)T^{2} \)
37 \( 1 + (1.68 + 0.439i)T + (0.873 + 0.487i)T^{2} \)
41 \( 1 + (0.660 + 0.750i)T^{2} \)
43 \( 1 + (-0.0800 + 0.0282i)T + (0.778 - 0.628i)T^{2} \)
47 \( 1 + (-0.996 - 0.0848i)T^{2} \)
53 \( 1 + (-0.721 - 0.691i)T^{2} \)
59 \( 1 + (0.911 - 0.411i)T^{2} \)
61 \( 1 + (1.03 + 0.914i)T + (0.127 + 0.991i)T^{2} \)
67 \( 1 + (-1.57 + 1.16i)T + (0.292 - 0.956i)T^{2} \)
71 \( 1 + (0.911 - 0.411i)T^{2} \)
73 \( 1 + (1.18 + 0.100i)T + (0.985 + 0.169i)T^{2} \)
79 \( 1 + (0.274 - 0.607i)T + (-0.660 - 0.750i)T^{2} \)
83 \( 1 + (0.778 + 0.628i)T^{2} \)
89 \( 1 + (0.942 - 0.333i)T^{2} \)
97 \( 1 + (-0.139 - 0.811i)T + (-0.942 + 0.333i)T^{2} \)
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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

−9.450868618424204731002675805554, −8.554220902753800529830286583123, −8.037457068018739059818304848658, −7.11431529427096674339116512055, −6.49207827442953217449454494157, −5.63269249389820651536293373708, −4.59184471108256051187146005956, −3.46227359819277342903725674867, −2.88520714711199776153378542730, −1.70599389400567948928978447481, 0.891942139725115046221059192905, 2.27393850831489267889806126760, 3.11923843451204953874455994040, 4.44738055058424177267759552589, 5.13678103117325859653240785259, 6.17392252188009582052948791382, 6.80283299326078884402933681240, 7.34824637598835368786450966536, 8.508937669727244330402819770068, 9.260237535485012473232802112697

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