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Properties

Label 2-2007-223.125-c0-0-0
Degree $2$
Conductor $2007$
Sign $0.292 - 0.956i$
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  + (0.996 − 0.0848i)4-s + (−0.415 + 1.92i)7-s + (−1.59 + 1.17i)13-s + (0.985 − 0.169i)16-s + (1.08 − 0.489i)19-s + (−0.967 − 0.251i)25-s + (−0.250 + 1.95i)28-s + (−0.492 + 1.22i)31-s + (−0.617 + 0.417i)37-s + (1.76 − 0.459i)43-s + (−2.62 − 1.18i)49-s + (−1.48 + 1.31i)52-s + (1.57 − 1.16i)61-s + (0.967 − 0.251i)64-s + (−0.144 − 1.69i)67-s + ⋯
L(s)  = 1  + (0.996 − 0.0848i)4-s + (−0.415 + 1.92i)7-s + (−1.59 + 1.17i)13-s + (0.985 − 0.169i)16-s + (1.08 − 0.489i)19-s + (−0.967 − 0.251i)25-s + (−0.250 + 1.95i)28-s + (−0.492 + 1.22i)31-s + (−0.617 + 0.417i)37-s + (1.76 − 0.459i)43-s + (−2.62 − 1.18i)49-s + (−1.48 + 1.31i)52-s + (1.57 − 1.16i)61-s + (0.967 − 0.251i)64-s + (−0.144 − 1.69i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $0.292 - 0.956i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (1909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ 0.292 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.273508418\)
\(L(\frac12)\) \(\approx\) \(1.273508418\)
\(L(1)\) 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 \( 1 + (-0.967 - 0.251i)T \)
good2 \( 1 + (-0.996 + 0.0848i)T^{2} \)
5 \( 1 + (0.967 + 0.251i)T^{2} \)
7 \( 1 + (0.415 - 1.92i)T + (-0.911 - 0.411i)T^{2} \)
11 \( 1 + (-0.372 - 0.927i)T^{2} \)
13 \( 1 + (1.59 - 1.17i)T + (0.292 - 0.956i)T^{2} \)
17 \( 1 + (0.372 + 0.927i)T^{2} \)
19 \( 1 + (-1.08 + 0.489i)T + (0.660 - 0.750i)T^{2} \)
23 \( 1 + (0.967 + 0.251i)T^{2} \)
29 \( 1 + (0.942 - 0.333i)T^{2} \)
31 \( 1 + (0.492 - 1.22i)T + (-0.721 - 0.691i)T^{2} \)
37 \( 1 + (0.617 - 0.417i)T + (0.372 - 0.927i)T^{2} \)
41 \( 1 + (-0.594 + 0.803i)T^{2} \)
43 \( 1 + (-1.76 + 0.459i)T + (0.873 - 0.487i)T^{2} \)
47 \( 1 + (0.660 + 0.750i)T^{2} \)
53 \( 1 + (0.210 + 0.977i)T^{2} \)
59 \( 1 + (0.450 - 0.892i)T^{2} \)
61 \( 1 + (-1.57 + 1.16i)T + (0.292 - 0.956i)T^{2} \)
67 \( 1 + (0.144 + 1.69i)T + (-0.985 + 0.169i)T^{2} \)
71 \( 1 + (0.450 - 0.892i)T^{2} \)
73 \( 1 + (-1.31 - 1.49i)T + (-0.127 + 0.991i)T^{2} \)
79 \( 1 + (-0.449 + 0.226i)T + (0.594 - 0.803i)T^{2} \)
83 \( 1 + (0.873 + 0.487i)T^{2} \)
89 \( 1 + (-0.967 + 0.251i)T^{2} \)
97 \( 1 + (-1.77 + 0.226i)T + (0.967 - 0.251i)T^{2} \)
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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

−9.470390959014982259298495036481, −8.899237956666546339686883504440, −7.84990019689327699047248257551, −7.04097078417765238510322993063, −6.41717414968982571557108757594, −5.50838381573878251045410426875, −4.96771021572825177043060422915, −3.41568675646620607673864801894, −2.47152890762937780566175119585, −1.98135799415829042632440415346, 0.877566988816937075755073658808, 2.32874146721963714523225132117, 3.35107510913264793037947083569, 4.05996693566558320126054012781, 5.27275484322929566573372690965, 6.07965675253188061280262724638, 7.20206376851112610061737573311, 7.45729978970240345054397393750, 7.927065969105991513942370571426, 9.555482250984508356721902742942

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