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Properties

Label 2-1001-7.2-c1-0-26
Degree $2$
Conductor $1001$
Sign $0.720 + 0.693i$
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  + (−1.15 + 2.00i)2-s + (−1.60 − 2.77i)3-s + (−1.68 − 2.92i)4-s + (−1.10 + 1.91i)5-s + 7.43·6-s + (−0.570 − 2.58i)7-s + 3.18·8-s + (−3.64 + 6.30i)9-s + (−2.55 − 4.42i)10-s + (0.5 + 0.866i)11-s + (−5.40 + 9.36i)12-s + 13-s + (5.84 + 1.85i)14-s + 7.07·15-s + (−0.316 + 0.548i)16-s + (3.16 + 5.47i)17-s + ⋯
L(s)  = 1  + (−0.819 + 1.41i)2-s + (−0.925 − 1.60i)3-s + (−0.843 − 1.46i)4-s + (−0.493 + 0.854i)5-s + 3.03·6-s + (−0.215 − 0.976i)7-s + 1.12·8-s + (−1.21 + 2.10i)9-s + (−0.808 − 1.40i)10-s + (0.150 + 0.261i)11-s + (−1.56 + 2.70i)12-s + 0.277·13-s + (1.56 + 0.494i)14-s + 1.82·15-s + (−0.0791 + 0.137i)16-s + (0.766 + 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.720 + 0.693i$
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: \((2,\ 1001,\ (\ :1/2),\ 0.720 + 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3656655002\)
\(L(\frac12)\) \(\approx\) \(0.3656655002\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.570 + 2.58i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 - T \)
good2 \( 1 + (1.15 - 2.00i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.60 + 2.77i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.10 - 1.91i)T + (-2.5 - 4.33i)T^{2} \)
17 \( 1 + (-3.16 - 5.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.92 - 3.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.515 + 0.892i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.02T + 29T^{2} \)
31 \( 1 + (4.22 + 7.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.83 + 4.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 + 3.65T + 43T^{2} \)
47 \( 1 + (-1.90 + 3.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.03 + 5.26i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.61 + 4.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.05 + 7.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.41 - 4.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.77T + 71T^{2} \)
73 \( 1 + (-2.50 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.56 + 14.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + (-2.30 + 3.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.39T + 97T^{2} \)
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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

−9.927916357363499945566890450164, −8.404353209310966592428066702728, −7.85375352814722561706535982852, −7.32038674472613561916154334744, −6.54539607700662384687293055183, −6.27156328008921559721263215083, −5.24283833465735396512242584669, −3.63922703912976561007105149223, −1.68386857833050055460474475700, −0.35656762748389227334920239651, 0.892762396289623500683935499490, 2.87223430406585765688652905963, 3.61359559179856432478098570773, 4.76225766887368324656300062829, 5.26253751488826870137847123181, 6.49737651802947155082203339983, 8.317455620989027733164931240473, 8.886187016713606924153578627745, 9.422868547437409404984904661483, 10.06783123831733140062598000111

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