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Properties

Label 2-2007-223.219-c0-0-0
Degree $2$
Conductor $2007$
Sign $0.994 - 0.101i$
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  + (−0.778 + 0.628i)4-s + (0.0535 − 0.417i)7-s + (1.58 − 0.634i)13-s + (0.210 − 0.977i)16-s + (−0.721 + 0.187i)19-s + (−0.450 + 0.892i)25-s + (0.220 + 0.358i)28-s + (1.73 − 0.148i)31-s + (−0.0845 − 1.99i)37-s + (0.871 + 1.72i)43-s + (0.795 + 0.207i)49-s + (−0.830 + 1.48i)52-s + (1.84 − 0.739i)61-s + (0.450 + 0.892i)64-s + (1.20 + 1.48i)67-s + ⋯
L(s)  = 1  + (−0.778 + 0.628i)4-s + (0.0535 − 0.417i)7-s + (1.58 − 0.634i)13-s + (0.210 − 0.977i)16-s + (−0.721 + 0.187i)19-s + (−0.450 + 0.892i)25-s + (0.220 + 0.358i)28-s + (1.73 − 0.148i)31-s + (−0.0845 − 1.99i)37-s + (0.871 + 1.72i)43-s + (0.795 + 0.207i)49-s + (−0.830 + 1.48i)52-s + (1.84 − 0.739i)61-s + (0.450 + 0.892i)64-s + (1.20 + 1.48i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $0.994 - 0.101i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ 0.994 - 0.101i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024762650\)
\(L(\frac12)\) \(\approx\) \(1.024762650\)
\(L(1)\) 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 \( 1 + (-0.450 + 0.892i)T \)
good2 \( 1 + (0.778 - 0.628i)T^{2} \)
5 \( 1 + (0.450 - 0.892i)T^{2} \)
7 \( 1 + (-0.0535 + 0.417i)T + (-0.967 - 0.251i)T^{2} \)
11 \( 1 + (0.996 + 0.0848i)T^{2} \)
13 \( 1 + (-1.58 + 0.634i)T + (0.721 - 0.691i)T^{2} \)
17 \( 1 + (-0.996 - 0.0848i)T^{2} \)
19 \( 1 + (0.721 - 0.187i)T + (0.873 - 0.487i)T^{2} \)
23 \( 1 + (0.450 - 0.892i)T^{2} \)
29 \( 1 + (-0.911 - 0.411i)T^{2} \)
31 \( 1 + (-1.73 + 0.148i)T + (0.985 - 0.169i)T^{2} \)
37 \( 1 + (0.0845 + 1.99i)T + (-0.996 + 0.0848i)T^{2} \)
41 \( 1 + (0.372 - 0.927i)T^{2} \)
43 \( 1 + (-0.871 - 1.72i)T + (-0.594 + 0.803i)T^{2} \)
47 \( 1 + (0.873 + 0.487i)T^{2} \)
53 \( 1 + (-0.127 - 0.991i)T^{2} \)
59 \( 1 + (0.828 + 0.559i)T^{2} \)
61 \( 1 + (-1.84 + 0.739i)T + (0.721 - 0.691i)T^{2} \)
67 \( 1 + (-1.20 - 1.48i)T + (-0.210 + 0.977i)T^{2} \)
71 \( 1 + (0.828 + 0.559i)T^{2} \)
73 \( 1 + (1.35 + 0.758i)T + (0.524 + 0.851i)T^{2} \)
79 \( 1 + (-1.00 - 1.47i)T + (-0.372 + 0.927i)T^{2} \)
83 \( 1 + (-0.594 - 0.803i)T^{2} \)
89 \( 1 + (-0.450 - 0.892i)T^{2} \)
97 \( 1 + (0.953 + 0.587i)T + (0.450 + 0.892i)T^{2} \)
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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

−9.270403708988208481748489875007, −8.434362749590871005450525651777, −8.034163892938510816978785129283, −7.13787412127811678114319441889, −6.11994919456895610298032867672, −5.36267821132021024830196912395, −4.17825406360851386751525010650, −3.79005804489976550670090731397, −2.68197961154402279782594822692, −1.03163140141231452009546031784, 1.11143815179980708532091534824, 2.34004415102417895951788974289, 3.74986925696054619292182233474, 4.41709742450241741883318429198, 5.33377197156117833641749586055, 6.22254092265139548581209558236, 6.64996247596368786245427362355, 8.170847575557064701417626223711, 8.565756075803384166776561733501, 9.201983078653072553709861954924

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