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Properties

Label 1-2009-2009.3-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.747 + 0.664i$
Analytic cond. $9.32975$
Root an. cond. $9.32975$
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  + (−0.680 + 0.733i)2-s + (−0.593 − 0.804i)3-s + (−0.0747 − 0.997i)4-s + (0.930 − 0.365i)5-s + (0.993 + 0.111i)6-s + (0.781 + 0.623i)8-s + (−0.294 + 0.955i)9-s + (−0.365 + 0.930i)10-s + (0.884 + 0.467i)11-s + (−0.757 + 0.652i)12-s + (−0.846 + 0.532i)13-s + (−0.846 − 0.532i)15-s + (−0.988 + 0.149i)16-s + (0.982 − 0.185i)17-s + (−0.5 − 0.866i)18-s + (0.258 + 0.965i)19-s + ⋯
L(s)  = 1  + (−0.680 + 0.733i)2-s + (−0.593 − 0.804i)3-s + (−0.0747 − 0.997i)4-s + (0.930 − 0.365i)5-s + (0.993 + 0.111i)6-s + (0.781 + 0.623i)8-s + (−0.294 + 0.955i)9-s + (−0.365 + 0.930i)10-s + (0.884 + 0.467i)11-s + (−0.757 + 0.652i)12-s + (−0.846 + 0.532i)13-s + (−0.846 − 0.532i)15-s + (−0.988 + 0.149i)16-s + (0.982 − 0.185i)17-s + (−0.5 − 0.866i)18-s + (0.258 + 0.965i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.747 + 0.664i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9988944469 + 0.3795854864i\)
\(L(\frac12)\) \(\approx\) \(0.9988944469 + 0.3795854864i\)
\(L(1)\) \(\approx\) \(0.7770538919 + 0.09453134074i\)
\(L(1)\) \(\approx\) \(0.7770538919 + 0.09453134074i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.680 + 0.733i)T \)
3 \( 1 + (-0.593 - 0.804i)T \)
5 \( 1 + (0.930 - 0.365i)T \)
11 \( 1 + (0.884 + 0.467i)T \)
13 \( 1 + (-0.846 + 0.532i)T \)
17 \( 1 + (0.982 - 0.185i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (-0.826 + 0.563i)T \)
29 \( 1 + (0.943 + 0.330i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.0747 - 0.997i)T \)
43 \( 1 + (0.781 - 0.623i)T \)
47 \( 1 + (-0.999 + 0.0373i)T \)
53 \( 1 + (-0.652 + 0.757i)T \)
59 \( 1 + (-0.365 + 0.930i)T \)
61 \( 1 + (0.997 + 0.0747i)T \)
67 \( 1 + (0.965 + 0.258i)T \)
71 \( 1 + (0.330 + 0.943i)T \)
73 \( 1 + (-0.680 - 0.733i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (0.467 + 0.884i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.91521765380479811274285969917, −19.16101621027407329526018477594, −18.250750738947984482225652672536, −17.54486656731837747129859019068, −17.23006943149492301953676651975, −16.45133940254641209411023495741, −15.77479061748519346566859343097, −14.57511092857868128460557135015, −14.151377818706687505277499913629, −12.997442964169083536689249297191, −12.25610126771428750155280493746, −11.54685745788165570657404917873, −10.85165033163199485093249856851, −10.005030065713715355013671725723, −9.79069126258468406957734442898, −8.94593231332536585131450915519, −8.09432893439233925357341173880, −6.893896669872197488673835185308, −6.27118722932450863531772904557, −5.23980780256264735836046029777, −4.479089575846334520506071292437, −3.33613706795881944887603703773, −2.85630152188489824598924482868, −1.63477235635092453627878775758, −0.6188786517433702958719265889, 0.97227981247118466373403250918, 1.6614829392369289601177470296, 2.37676883706067102918616525799, 4.16861442532827758224612154857, 5.187210937806850018544557377430, 5.74629162751565084438988885803, 6.413427385094992387356175345376, 7.21148645470528900402589735707, 7.80963700370683832718457300232, 8.76536454486610970366702759839, 9.71232718857646866493064463583, 9.98263173502012035188700107458, 11.08833336818506703426682689526, 12.06189593583303303431843503583, 12.51079579585285438048345230047, 13.68122249655831909322771985343, 14.22442777097547226180162204905, 14.69441373755044963947991636661, 16.14045194916786135068825997563, 16.52618934706834096068465798768, 17.28140828270887379743270979953, 17.63057355280948981715090586422, 18.40911933727762588492070274362, 19.07546623281762302111906125061, 19.775452097468897545929267538400

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