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Properties

Label 2-2007-223.91-c0-0-0
Degree $2$
Conductor $2007$
Sign $0.929 + 0.368i$
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  + (0.967 − 0.251i)4-s + (1.03 − 1.40i)7-s + (0.618 + 1.74i)13-s + (0.873 − 0.487i)16-s + (−0.552 + 1.80i)19-s + (−0.721 − 0.691i)25-s + (0.651 − 1.62i)28-s + (−1.50 + 0.682i)31-s + (−0.383 − 1.78i)37-s + (0.422 − 0.405i)43-s + (−0.599 − 1.95i)49-s + (1.03 + 1.53i)52-s + (−0.535 − 1.51i)61-s + (0.721 − 0.691i)64-s + (−0.0427 − 0.164i)67-s + ⋯
L(s)  = 1  + (0.967 − 0.251i)4-s + (1.03 − 1.40i)7-s + (0.618 + 1.74i)13-s + (0.873 − 0.487i)16-s + (−0.552 + 1.80i)19-s + (−0.721 − 0.691i)25-s + (0.651 − 1.62i)28-s + (−1.50 + 0.682i)31-s + (−0.383 − 1.78i)37-s + (0.422 − 0.405i)43-s + (−0.599 − 1.95i)49-s + (1.03 + 1.53i)52-s + (−0.535 − 1.51i)61-s + (0.721 − 0.691i)64-s + (−0.0427 − 0.164i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $0.929 + 0.368i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ 0.929 + 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.631314040\)
\(L(\frac12)\) \(\approx\) \(1.631314040\)
\(L(1)\) 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 \( 1 + (-0.721 - 0.691i)T \)
good2 \( 1 + (-0.967 + 0.251i)T^{2} \)
5 \( 1 + (0.721 + 0.691i)T^{2} \)
7 \( 1 + (-1.03 + 1.40i)T + (-0.292 - 0.956i)T^{2} \)
11 \( 1 + (0.911 + 0.411i)T^{2} \)
13 \( 1 + (-0.618 - 1.74i)T + (-0.778 + 0.628i)T^{2} \)
17 \( 1 + (-0.911 - 0.411i)T^{2} \)
19 \( 1 + (0.552 - 1.80i)T + (-0.828 - 0.559i)T^{2} \)
23 \( 1 + (0.721 + 0.691i)T^{2} \)
29 \( 1 + (0.524 - 0.851i)T^{2} \)
31 \( 1 + (1.50 - 0.682i)T + (0.660 - 0.750i)T^{2} \)
37 \( 1 + (0.383 + 1.78i)T + (-0.911 + 0.411i)T^{2} \)
41 \( 1 + (0.942 + 0.333i)T^{2} \)
43 \( 1 + (-0.422 + 0.405i)T + (0.0424 - 0.999i)T^{2} \)
47 \( 1 + (-0.828 + 0.559i)T^{2} \)
53 \( 1 + (-0.594 - 0.803i)T^{2} \)
59 \( 1 + (-0.985 + 0.169i)T^{2} \)
61 \( 1 + (0.535 + 1.51i)T + (-0.778 + 0.628i)T^{2} \)
67 \( 1 + (0.0427 + 0.164i)T + (-0.873 + 0.487i)T^{2} \)
71 \( 1 + (-0.985 + 0.169i)T^{2} \)
73 \( 1 + (1.60 - 1.08i)T + (0.372 - 0.927i)T^{2} \)
79 \( 1 + (0.233 - 1.36i)T + (-0.942 - 0.333i)T^{2} \)
83 \( 1 + (0.0424 + 0.999i)T^{2} \)
89 \( 1 + (-0.721 + 0.691i)T^{2} \)
97 \( 1 + (-0.313 + 0.126i)T + (0.721 - 0.691i)T^{2} \)
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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

−9.346957272301795147959061024470, −8.344588591320996515972746566213, −7.60779732633387171715467490418, −7.02142798097209546667982802030, −6.26361669676510387647193623254, −5.38500071056760057426510388796, −4.11102476783689983449242227831, −3.79559069606373491017874558692, −1.97731468967273498374776291985, −1.51917932831136017325490968193, 1.58174702948797284333137124161, 2.57717360227309878994593896509, 3.24129826973426764480747973674, 4.67745549125508362490258239153, 5.60961788285457376580980140273, 6.00787745199879054678203082992, 7.18008935435602466163661528388, 7.87442803324983747879580758563, 8.528595510101942999415176897387, 9.164985458433727802005655749552

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