↑

Properties

Label 1-680-680.587-r0-0-0
Degree $1$
Conductor $680$
Sign $0.564 - 0.825i$
Analytic cond. $3.15790$
Root an. cond. $3.15790$
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  + (0.707 − 0.707i)3-s + (0.707 + 0.707i)7-s − i·9-s + (0.707 − 0.707i)11-s − i·13-s + i·19-s + 21-s + (0.707 + 0.707i)23-s + (−0.707 − 0.707i)27-s + (−0.707 − 0.707i)29-s + (−0.707 − 0.707i)31-s − i·33-s + (0.707 − 0.707i)37-s + (−0.707 − 0.707i)39-s + (−0.707 + 0.707i)41-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (0.707 + 0.707i)7-s − i·9-s + (0.707 − 0.707i)11-s − i·13-s + i·19-s + 21-s + (0.707 + 0.707i)23-s + (−0.707 − 0.707i)27-s + (−0.707 − 0.707i)29-s + (−0.707 − 0.707i)31-s − i·33-s + (0.707 − 0.707i)37-s + (−0.707 − 0.707i)39-s + (−0.707 + 0.707i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(3.15790\)
Root analytic conductor: \(3.15790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 680,\ (0:\ ),\ 0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.809411638 - 0.9543282547i\)
\(L(\frac12)\) \(\approx\) \(1.809411638 - 0.9543282547i\)
\(L(1)\) \(\approx\) \(1.424106181 - 0.4025996502i\)
\(L(1)\) \(\approx\) \(1.424106181 - 0.4025996502i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 - iT \)
19 \( 1 + iT \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + T \)
47 \( 1 - iT \)
53 \( 1 + T \)
59 \( 1 - iT \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + T \)
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

−22.72267180065815144385605668069, −21.89521143521862238867371270904, −21.181284281230577727014599377395, −20.34019312633147841777660481540, −19.87221716644958674144198062379, −18.94074130257173124653938186825, −17.884209953442823814465614017246, −16.89756761645819009356124065573, −16.4190442743259816403040486991, −15.16507434908636343445346805760, −14.65439395790294386209675231504, −13.92230826843447010744403618576, −13.09792905606959415869251427086, −11.82095810548263246157669174178, −10.9808256130885753210044353222, −10.22098076644824289992108410086, −9.1435025633020552991361267542, −8.70036153947009079873912225376, −7.38509535620992576085859937595, −6.85658595983827334934119026290, −5.17319371007825785986332662974, −4.4344096500816258055723220331, −3.74250058574302464252050715711, −2.43272917815265278013874037526, −1.42958386536690474258121719637, 1.04477343671452807210959399385, 2.04830605385617402770962754432, 3.089923970784832133055245452822, 4.00560921531938509613222364851, 5.53232236039583799491496012885, 6.121470278918336553693172826429, 7.46823664871464463011188549741, 8.06010147640169449150067900527, 8.8995153738192737053896535730, 9.67168943399954287892946946403, 11.06851509924264129837285543636, 11.77303976028575209621546265494, 12.705325310271392588531680152863, 13.412116482001346133268493028740, 14.511584988878100210559829636828, 14.83901423337132772118371136082, 15.85657955733458957286366321133, 17.07896396266266489049399602607, 17.80218217208029380404293603112, 18.681911442818088975443860814001, 19.17535178530898115419850717065, 20.197396724593316834449140631499, 20.84079380561629345065828839495, 21.69360794266771169951139397852, 22.62792861406981818326236245031

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