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Properties

Label 2-2007-223.118-c0-0-0
Degree $2$
Conductor $2007$
Sign $0.402 + 0.915i$
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  + (−0.0424 − 0.999i)4-s + (1.55 − 1.25i)7-s + (1.34 + 0.675i)13-s + (−0.996 + 0.0848i)16-s + (−0.189 − 0.880i)19-s + (−0.127 + 0.991i)25-s + (−1.31 − 1.49i)28-s + (−1.50 + 1.02i)31-s + (0.485 + 1.58i)37-s + (−0.0535 − 0.417i)43-s + (0.625 − 2.90i)49-s + (0.618 − 1.36i)52-s + (−1.12 − 0.565i)61-s + (0.127 + 0.991i)64-s + (−0.974 + 0.0413i)67-s + ⋯
L(s)  = 1  + (−0.0424 − 0.999i)4-s + (1.55 − 1.25i)7-s + (1.34 + 0.675i)13-s + (−0.996 + 0.0848i)16-s + (−0.189 − 0.880i)19-s + (−0.127 + 0.991i)25-s + (−1.31 − 1.49i)28-s + (−1.50 + 1.02i)31-s + (0.485 + 1.58i)37-s + (−0.0535 − 0.417i)43-s + (0.625 − 2.90i)49-s + (0.618 − 1.36i)52-s + (−1.12 − 0.565i)61-s + (0.127 + 0.991i)64-s + (−0.974 + 0.0413i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ 0.402 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.394821466\)
\(L(\frac12)\) \(\approx\) \(1.394821466\)
\(L(1)\) 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 \( 1 + (-0.127 + 0.991i)T \)
good2 \( 1 + (0.0424 + 0.999i)T^{2} \)
5 \( 1 + (0.127 - 0.991i)T^{2} \)
7 \( 1 + (-1.55 + 1.25i)T + (0.210 - 0.977i)T^{2} \)
11 \( 1 + (0.828 + 0.559i)T^{2} \)
13 \( 1 + (-1.34 - 0.675i)T + (0.594 + 0.803i)T^{2} \)
17 \( 1 + (-0.828 - 0.559i)T^{2} \)
19 \( 1 + (0.189 + 0.880i)T + (-0.911 + 0.411i)T^{2} \)
23 \( 1 + (0.127 - 0.991i)T^{2} \)
29 \( 1 + (0.985 - 0.169i)T^{2} \)
31 \( 1 + (1.50 - 1.02i)T + (0.372 - 0.927i)T^{2} \)
37 \( 1 + (-0.485 - 1.58i)T + (-0.828 + 0.559i)T^{2} \)
41 \( 1 + (-0.450 - 0.892i)T^{2} \)
43 \( 1 + (0.0535 + 0.417i)T + (-0.967 + 0.251i)T^{2} \)
47 \( 1 + (-0.911 - 0.411i)T^{2} \)
53 \( 1 + (0.778 + 0.628i)T^{2} \)
59 \( 1 + (-0.524 - 0.851i)T^{2} \)
61 \( 1 + (1.12 + 0.565i)T + (0.594 + 0.803i)T^{2} \)
67 \( 1 + (0.974 - 0.0413i)T + (0.996 - 0.0848i)T^{2} \)
71 \( 1 + (-0.524 - 0.851i)T^{2} \)
73 \( 1 + (-0.0773 - 0.0349i)T + (0.660 + 0.750i)T^{2} \)
79 \( 1 + (1.68 + 1.04i)T + (0.450 + 0.892i)T^{2} \)
83 \( 1 + (-0.967 - 0.251i)T^{2} \)
89 \( 1 + (-0.127 - 0.991i)T^{2} \)
97 \( 1 + (-1.27 - 1.12i)T + (0.127 + 0.991i)T^{2} \)
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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

−9.068857106108999510923693420467, −8.582300592519058505801961626254, −7.52289581988901235313881767184, −6.91154294442541758660934395581, −6.01374367539357477197967282846, −5.00445972329894605901652114881, −4.51540569720162172119572132417, −3.53909376034639837738536576471, −1.78224466073393431156563757341, −1.21475538128624105507467608818, 1.72141402641274041161509683415, 2.61778882545382034985114514874, 3.75958230618146120457421884195, 4.51358420338974289929237172470, 5.62821747091940659238855960602, 6.08184635064342355780389801835, 7.58320809777270199415504414990, 7.907644017602843813765439469785, 8.725730000254804162204991144558, 9.014030902686427285129526710516

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