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Properties

Label 2-1001-1001.802-c0-0-1
Degree $2$
Conductor $1001$
Sign $0.190 + 0.981i$
Analytic cond. $0.499564$
Root an. cond. $0.706798$
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  − i·2-s + (0.5 + 0.866i)5-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)10-s + (0.866 − 0.5i)11-s + i·13-s + (−0.5 + 0.866i)14-s − 16-s + i·17-s + (−0.866 − 0.5i)18-s + (−0.5 − 0.866i)22-s − 23-s + 26-s + ⋯
L(s)  = 1  − i·2-s + (0.5 + 0.866i)5-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)10-s + (0.866 − 0.5i)11-s + i·13-s + (−0.5 + 0.866i)14-s − 16-s + i·17-s + (−0.866 − 0.5i)18-s + (−0.5 − 0.866i)22-s − 23-s + 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.190 + 0.981i$
Analytic conductor: \(0.499564\)
Root analytic conductor: \(0.706798\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :0),\ 0.190 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.197329505\)
\(L(\frac12)\) \(\approx\) \(1.197329505\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 - iT \)
good2 \( 1 + iT - T^{2} \)
3 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.10268898330986404574930725519, −9.583820706902927956372707337428, −8.666917817016221326842446111593, −7.07908667732282814865493160112, −6.57720785315680201397726664817, −6.12542677456107271019254692501, −4.01444386168600875334206217982, −3.70670595174204676369529199153, −2.55536619408814098986258720507, −1.36140990295399330555221714244, 1.76515573163860074478116502237, 2.99933385952344453517206981419, 4.67898749166580317124707299478, 5.26349385205020840835457148236, 6.19303479585722649545549836645, 6.86775207501772776660947997061, 7.83772097890792156652404687558, 8.528052873965925705604008961977, 9.473043898576265222285982335485, 10.03524121456205298383667434224

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