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Properties

Label 2-2007-223.221-c0-0-0
Degree $2$
Conductor $2007$
Sign $0.00798 - 0.999i$
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  + (−0.942 + 0.333i)4-s + (−1.02 − 1.16i)7-s + (−0.546 + 0.808i)13-s + (0.778 − 0.628i)16-s + (0.210 + 1.64i)19-s + (0.524 + 0.851i)25-s + (1.35 + 0.758i)28-s + (0.0821 + 1.93i)31-s + (−0.0612 − 0.0587i)37-s + (−0.133 + 0.216i)43-s + (−0.180 + 1.40i)49-s + (0.245 − 0.943i)52-s + (−0.840 + 1.24i)61-s + (−0.524 + 0.851i)64-s + (−0.535 − 1.51i)67-s + ⋯
L(s)  = 1  + (−0.942 + 0.333i)4-s + (−1.02 − 1.16i)7-s + (−0.546 + 0.808i)13-s + (0.778 − 0.628i)16-s + (0.210 + 1.64i)19-s + (0.524 + 0.851i)25-s + (1.35 + 0.758i)28-s + (0.0821 + 1.93i)31-s + (−0.0612 − 0.0587i)37-s + (−0.133 + 0.216i)43-s + (−0.180 + 1.40i)49-s + (0.245 − 0.943i)52-s + (−0.840 + 1.24i)61-s + (−0.524 + 0.851i)64-s + (−0.535 − 1.51i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00798 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00798 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $0.00798 - 0.999i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ 0.00798 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5595490710\)
\(L(\frac12)\) \(\approx\) \(0.5595490710\)
\(L(1)\) 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 \( 1 + (0.524 + 0.851i)T \)
good2 \( 1 + (0.942 - 0.333i)T^{2} \)
5 \( 1 + (-0.524 - 0.851i)T^{2} \)
7 \( 1 + (1.02 + 1.16i)T + (-0.127 + 0.991i)T^{2} \)
11 \( 1 + (-0.0424 + 0.999i)T^{2} \)
13 \( 1 + (0.546 - 0.808i)T + (-0.372 - 0.927i)T^{2} \)
17 \( 1 + (0.0424 - 0.999i)T^{2} \)
19 \( 1 + (-0.210 - 1.64i)T + (-0.967 + 0.251i)T^{2} \)
23 \( 1 + (-0.524 - 0.851i)T^{2} \)
29 \( 1 + (0.210 - 0.977i)T^{2} \)
31 \( 1 + (-0.0821 - 1.93i)T + (-0.996 + 0.0848i)T^{2} \)
37 \( 1 + (0.0612 + 0.0587i)T + (0.0424 + 0.999i)T^{2} \)
41 \( 1 + (-0.828 + 0.559i)T^{2} \)
43 \( 1 + (0.133 - 0.216i)T + (-0.450 - 0.892i)T^{2} \)
47 \( 1 + (-0.967 - 0.251i)T^{2} \)
53 \( 1 + (0.660 - 0.750i)T^{2} \)
59 \( 1 + (0.292 - 0.956i)T^{2} \)
61 \( 1 + (0.840 - 1.24i)T + (-0.372 - 0.927i)T^{2} \)
67 \( 1 + (0.535 + 1.51i)T + (-0.778 + 0.628i)T^{2} \)
71 \( 1 + (0.292 - 0.956i)T^{2} \)
73 \( 1 + (-1.82 - 0.475i)T + (0.873 + 0.487i)T^{2} \)
79 \( 1 + (1.62 - 0.498i)T + (0.828 - 0.559i)T^{2} \)
83 \( 1 + (-0.450 + 0.892i)T^{2} \)
89 \( 1 + (0.524 - 0.851i)T^{2} \)
97 \( 1 + (-0.932 - 1.66i)T + (-0.524 + 0.851i)T^{2} \)
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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

−9.540379010031034007316672109810, −8.888399964417550749710661011719, −7.942994173896875547238313076444, −7.21918659845370524111939215008, −6.54624829803794474109527062191, −5.44118254777561788776970150998, −4.53353934485589583001327383097, −3.74453265025269528653576487741, −3.13355041397079555205697336387, −1.33485361547724191194855750065, 0.44426751457239024914772514301, 2.40983279720896092797920752698, 3.15840556988416771559363638889, 4.36961238779262546904280327192, 5.18473544133686485579426099094, 5.87521687779324462853907386350, 6.62178356325162335886288894185, 7.72059320914813333479962611045, 8.594808753967054296438786658930, 9.218221527571065275216661398645

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