↑

Properties

Label 2-1071-17.13-c1-0-44
Degree $2$
Conductor $1071$
Sign $-0.351 - 0.936i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.20i·2-s + 0.539·4-s + (−1.52 − 1.52i)5-s + (0.707 − 0.707i)7-s − 3.06i·8-s + (−1.84 + 1.84i)10-s + (−4.46 + 4.46i)11-s − 5.08·13-s + (−0.854 − 0.854i)14-s − 2.63·16-s + (−4.00 + 0.968i)17-s + 7.96i·19-s + (−0.824 − 0.824i)20-s + (5.39 + 5.39i)22-s + (1.06 − 1.06i)23-s + ⋯
L(s)  = 1  − 0.854i·2-s + 0.269·4-s + (−0.683 − 0.683i)5-s + (0.267 − 0.267i)7-s − 1.08i·8-s + (−0.584 + 0.584i)10-s + (−1.34 + 1.34i)11-s − 1.41·13-s + (−0.228 − 0.228i)14-s − 0.657·16-s + (−0.972 + 0.234i)17-s + 1.82i·19-s + (−0.184 − 0.184i)20-s + (1.15 + 1.15i)22-s + (0.222 − 0.222i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1071\)    =    \(3^{2} \cdot 7 \cdot 17\)
Sign: $-0.351 - 0.936i$
Analytic conductor: \(8.55197\)
Root analytic conductor: \(2.92437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1071} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1071,\ (\ :1/2),\ -0.351 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1387436535\)
\(L(\frac12)\) \(\approx\) \(0.1387436535\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (4.00 - 0.968i)T \)
good2 \( 1 + 1.20iT - 2T^{2} \)
5 \( 1 + (1.52 + 1.52i)T + 5iT^{2} \)
11 \( 1 + (4.46 - 4.46i)T - 11iT^{2} \)
13 \( 1 + 5.08T + 13T^{2} \)
19 \( 1 - 7.96iT - 19T^{2} \)
23 \( 1 + (-1.06 + 1.06i)T - 23iT^{2} \)
29 \( 1 + (2.62 + 2.62i)T + 29iT^{2} \)
31 \( 1 + (-3.12 - 3.12i)T + 31iT^{2} \)
37 \( 1 + (5.76 + 5.76i)T + 37iT^{2} \)
41 \( 1 + (-5.51 + 5.51i)T - 41iT^{2} \)
43 \( 1 + 7.20iT - 43T^{2} \)
47 \( 1 - 1.30T + 47T^{2} \)
53 \( 1 - 6.85iT - 53T^{2} \)
59 \( 1 - 12.2iT - 59T^{2} \)
61 \( 1 + (3.53 - 3.53i)T - 61iT^{2} \)
67 \( 1 + 0.226T + 67T^{2} \)
71 \( 1 + (0.416 + 0.416i)T + 71iT^{2} \)
73 \( 1 + (9.39 + 9.39i)T + 73iT^{2} \)
79 \( 1 + (-3.12 + 3.12i)T - 79iT^{2} \)
83 \( 1 - 2.55iT - 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + (9.49 + 9.49i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.604811713874592006931906790000, −8.503087861023808160303157012846, −7.45428097710188079662185204990, −7.25499000234051819499481049868, −5.70591271182332653316135476610, −4.60471077071146526913376623896, −4.05319567563295553225301538492, −2.60012915769198763672759194912, −1.81309400592031309888761839944, −0.05478373852315421997190796934, 2.51935236231016852566843000166, 2.99842052891849737840760913209, 4.77684825051265010539799957202, 5.32173498096776185228665281227, 6.48654981993335073677894737740, 7.13476744042061179314583129909, 7.83227432138008737307658542898, 8.464903073356388038559554644820, 9.477838135029890032659920925813, 10.74144150884940168981157280778

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