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Properties

Label 2-1001-1001.835-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: $\chi_{1001} (835, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :0),\ -0.190 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.293168199\)
\(L(\frac12)\) \(\approx\) \(1.293168199\)
\(L(1)\) 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

−10.24021226267681300241131164361, −9.301277450781665994110401951615, −8.355176190808437733761683496884, −7.61536945652701289332391028836, −6.55574390315962552059619198822, −5.32503203762855870038750783181, −4.58261478765421054650966746777, −3.60036354586455519294944482227, −2.01096082109414896127623069058, −1.54909303729338259450901064532, 2.07860450170559186856399060097, 2.95294577949766193018173481421, 4.58403597006240096911894268710, 5.54774609450773310490712982960, 6.14307674087837444262381697494, 7.10884185537766579353457782642, 7.70478026222861143593788628056, 8.438199882763069376830005429985, 9.631528974919362617217289450912, 10.26821783715713611261867820572

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