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Properties

Label 2-1071-7.4-c1-0-12
Degree $2$
Conductor $1071$
Sign $-0.908 + 0.418i$
Analytic cond. $8.55197$
Root an. cond. $2.92437$
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  + (1.19 + 2.07i)2-s + (−1.87 + 3.23i)4-s + (2.18 + 3.77i)5-s + (−2.64 − 0.0756i)7-s − 4.17·8-s + (−5.22 + 9.05i)10-s + (−1.09 + 1.89i)11-s + 4.97·13-s + (−3.01 − 5.57i)14-s + (−1.25 − 2.17i)16-s + (0.5 − 0.866i)17-s + (−1.28 − 2.23i)19-s − 16.3·20-s − 5.25·22-s + (−2.98 − 5.16i)23-s + ⋯
L(s)  = 1  + (0.847 + 1.46i)2-s + (−0.935 + 1.61i)4-s + (0.975 + 1.69i)5-s + (−0.999 − 0.0285i)7-s − 1.47·8-s + (−1.65 + 2.86i)10-s + (−0.330 + 0.572i)11-s + 1.38·13-s + (−0.804 − 1.49i)14-s + (−0.314 − 0.544i)16-s + (0.121 − 0.210i)17-s + (−0.295 − 0.512i)19-s − 3.65·20-s − 1.12·22-s + (−0.622 − 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1071\)    =    \(3^{2} \cdot 7 \cdot 17\)
Sign: $-0.908 + 0.418i$
Analytic conductor: \(8.55197\)
Root analytic conductor: \(2.92437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1071} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1071,\ (\ :1/2),\ -0.908 + 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.528440685\)
\(L(\frac12)\) \(\approx\) \(2.528440685\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2.64 + 0.0756i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.19 - 2.07i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-2.18 - 3.77i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.09 - 1.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
19 \( 1 + (1.28 + 2.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.98 + 5.16i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.59T + 29T^{2} \)
31 \( 1 + (-1.62 + 2.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.41 + 5.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.32T + 41T^{2} \)
43 \( 1 - 3.15T + 43T^{2} \)
47 \( 1 + (-0.0918 - 0.159i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.42 - 2.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.78 - 6.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.96 + 5.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.801 - 1.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + (4.18 - 7.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.55 + 6.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.625T + 83T^{2} \)
89 \( 1 + (-0.479 - 0.830i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.43T + 97T^{2} \)
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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

−10.37664675390624285045032516724, −9.480625594234478286226362830338, −8.461707444319237581995204496350, −7.37602443680941361775856244616, −6.72786408663522145252219776577, −6.23003738333837799564151065607, −5.72063397988734903285576554071, −4.35773618350740029837154000750, −3.35781945356177379904653535774, −2.47036541806461291899129628623, 0.896860343532345154346116796773, 1.75843359018744641588835165986, 3.06848110023916950356929863683, 3.96063020197611097421081639756, 4.89831134025034272879850195672, 5.78338071524941949554101862203, 6.22939237461145638914261935125, 8.243695669960643487186757282494, 8.857871348029165180821958133821, 9.760189696106773211974329506338

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