↑

Properties

Label 1-287-287.68-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.718 - 0.695i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.258 − 0.965i)3-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.707 − 0.707i)6-s − i·8-s + (−0.866 + 0.5i)9-s + (0.5 − 0.866i)10-s + (−0.258 − 0.965i)11-s + (−0.965 − 0.258i)12-s + (−0.707 + 0.707i)13-s + (−0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (0.965 − 0.258i)17-s + (−0.5 + 0.866i)18-s + (−0.258 + 0.965i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.258 − 0.965i)3-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.707 − 0.707i)6-s − i·8-s + (−0.866 + 0.5i)9-s + (0.5 − 0.866i)10-s + (−0.258 − 0.965i)11-s + (−0.965 − 0.258i)12-s + (−0.707 + 0.707i)13-s + (−0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (0.965 − 0.258i)17-s + (−0.5 + 0.866i)18-s + (−0.258 + 0.965i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.718 - 0.695i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7578446325 - 1.870859353i\)
\(L(\frac12)\) \(\approx\) \(0.7578446325 - 1.870859353i\)
\(L(1)\) \(\approx\) \(1.203859283 - 1.161646420i\)
\(L(1)\) \(\approx\) \(1.203859283 - 1.161646420i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (0.965 - 0.258i)T \)
19 \( 1 + (-0.258 + 0.965i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.258 + 0.965i)T \)
53 \( 1 + (-0.258 - 0.965i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (-0.965 + 0.258i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (0.965 + 0.258i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.73396400465341305207791122282, −25.19134891225173693353861746601, −23.89956786895952907303750433160, −22.88847128266612935270000979818, −22.31754517408876308157797326605, −21.57096894069207145287471357771, −20.792617418030907769328236884003, −19.983738266613043936677740210965, −18.21833589936652228388399378499, −17.2471790347986762350152492859, −16.78303783133860804190175490734, −15.37786919727317142858458732549, −14.95453215091091640661255888684, −14.10490133248653070924459201848, −12.92919908874631297428765751688, −12.03564795398702395383124115599, −10.72208844345224860993238876953, −10.06958233493385839725846109840, −8.85085926907377322015360641138, −7.417797387808101125409945335238, −6.40080121694995005379537679911, −5.32293870592100688334462992449, −4.71352949740348771223182196457, −3.29004226635694730896764578859, −2.38931316363310585321636380782, 1.11733482946978880633887865356, 2.093810452274720316702598596510, 3.25969845168849991746184068694, 4.95468850593459757481249594263, 5.716941947710620115386919840087, 6.53532630437216012433983951189, 7.806685043364152494301331563486, 9.2084785766443411252520707218, 10.31126779414517915816172750620, 11.43296245264996964957873922424, 12.30998352132805537146237449569, 13.010125214911802451954785127642, 14.008134038578090087306954235251, 14.3692488750428941036849637632, 16.14477467265115643591643323780, 16.88404103244249217900846772976, 18.04853922638673863840920418863, 19.060971503236587389417908966694, 19.61477794055335195017579772843, 21.03705560600076471232078870287, 21.390343026827480638885578845448, 22.547912296542045898170044211116, 23.45593494232313358152354127897, 24.186518394518743066937617515695, 24.91231039566872331378126372402

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