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Properties

Label 2-1001-7.2-c1-0-78
Degree $2$
Conductor $1001$
Sign $0.374 - 0.927i$
Analytic cond. $7.99302$
Root an. cond. $2.82719$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.31 − 2.28i)2-s + (−0.776 − 1.34i)3-s + (−2.48 − 4.30i)4-s + (0.109 − 0.189i)5-s − 4.10·6-s + (−2.18 + 1.49i)7-s − 7.83·8-s + (0.292 − 0.506i)9-s + (−0.288 − 0.499i)10-s + (−0.5 − 0.866i)11-s + (−3.86 + 6.68i)12-s − 13-s + (0.529 + 6.96i)14-s − 0.339·15-s + (−5.37 + 9.30i)16-s + (−0.735 − 1.27i)17-s + ⋯
L(s)  = 1  + (0.933 − 1.61i)2-s + (−0.448 − 0.776i)3-s + (−1.24 − 2.15i)4-s + (0.0488 − 0.0845i)5-s − 1.67·6-s + (−0.825 + 0.564i)7-s − 2.77·8-s + (0.0975 − 0.168i)9-s + (−0.0911 − 0.157i)10-s + (−0.150 − 0.261i)11-s + (−1.11 + 1.93i)12-s − 0.277·13-s + (0.141 + 1.86i)14-s − 0.0876·15-s + (−1.34 + 2.32i)16-s + (−0.178 − 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(7.99302\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (716, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :1/2),\ 0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.023793575\)
\(L(\frac12)\) \(\approx\) \(1.023793575\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.18 - 1.49i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (-1.31 + 2.28i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.776 + 1.34i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.109 + 0.189i)T + (-2.5 - 4.33i)T^{2} \)
17 \( 1 + (0.735 + 1.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.28 - 2.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.52 + 4.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.55T + 29T^{2} \)
31 \( 1 + (-1.94 - 3.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.65 + 6.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.91T + 41T^{2} \)
43 \( 1 - 5.76T + 43T^{2} \)
47 \( 1 + (1.20 - 2.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.47 + 2.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.44 + 11.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.79 - 6.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.70 + 13.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + (1.89 + 3.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.05 + 3.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.94T + 83T^{2} \)
89 \( 1 + (-0.928 + 1.60i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 19.1T + 97T^{2} \)
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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

−9.409009062437177440026374950630, −9.048392401720184655086170635172, −7.43233623434968555918612452320, −6.29596669862466463208829117071, −5.75062607529604306871411503247, −4.74923515384776725175644591720, −3.62615785878959182031035560004, −2.74745139421818163324134005524, −1.68163729630263053652801699201, −0.35723178790101201102187840880, 2.96122653565415258273075928409, 4.13928311182443490532800103384, 4.55157006755199971551101971554, 5.58962663030306521337624051227, 6.23410310279149846024062920697, 7.20843162586409643338811145724, 7.67214032271031762681762629765, 8.892864402106335381633852551751, 9.666859395006999845764637719454, 10.52649489014909549445425132007

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