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Properties

Label 1-680-680.99-r0-0-0
Degree $1$
Conductor $680$
Sign $0.507 + 0.861i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 + 0.382i)3-s + (0.382 + 0.923i)7-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s − i·13-s + (0.707 − 0.707i)19-s − i·21-s + (−0.923 + 0.382i)23-s + (0.382 + 0.923i)27-s + (0.382 − 0.923i)29-s + (−0.923 − 0.382i)31-s + 33-s + (−0.923 − 0.382i)37-s + (−0.382 + 0.923i)39-s + (0.382 + 0.923i)41-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)3-s + (0.382 + 0.923i)7-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s − i·13-s + (0.707 − 0.707i)19-s − i·21-s + (−0.923 + 0.382i)23-s + (0.382 + 0.923i)27-s + (0.382 − 0.923i)29-s + (−0.923 − 0.382i)31-s + 33-s + (−0.923 − 0.382i)37-s + (−0.382 + 0.923i)39-s + (0.382 + 0.923i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.507 + 0.861i$
Analytic conductor: \(3.15790\)
Root analytic conductor: \(3.15790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 680,\ (0:\ ),\ 0.507 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.898812582 + 1.085159231i\)
\(L(\frac12)\) \(\approx\) \(1.898812582 + 1.085159231i\)
\(L(1)\) \(\approx\) \(1.504225021 + 0.4359191661i\)
\(L(1)\) \(\approx\) \(1.504225021 + 0.4359191661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 - iT \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (-0.923 + 0.382i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 + T \)
71 \( 1 + (0.923 + 0.382i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.382 - 0.923i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.67938652403644107910657409010, −21.72158455462695794133550197750, −20.61294783975787240242792858734, −20.11437193103012544667045879395, −19.66477336791960724666403324524, −18.4180198220760929012260273189, −17.85770361831430253778915156564, −16.917409639640546478852934477492, −15.939807187053406182297978827327, −14.90171292296075130289531328956, −14.25438458616154276606360923765, −13.66560941341118973654992445012, −12.61203546317272801485624021934, −11.95330714555306782379533821236, −10.608724985169121560657723736316, −9.94050926501760237414389100534, −8.91404027588460749380482729135, −8.0227437983278026209114162731, −7.323226184107394599188722060114, −6.49699492585795119391823017103, −5.14015768660608953191845669466, −3.92645658131525988397380198689, −3.34513546845489076342838796690, −1.93906179597532448041649873998, −1.06187134536562106210013781757, 1.58245879633643745886045194341, 2.42041813664042527841754523972, 3.55547282536182468811251642163, 4.402371177035344702472447572209, 5.46477970040943145226391651260, 6.59248386981995605198153385992, 7.65707868395384619335871624737, 8.62597537959503215301993553734, 9.20305527661276094856345386867, 9.92259192184355437461065173275, 11.31734929946912320679549608127, 11.81731226075914478568849163340, 13.01462764584449136750832572147, 14.06805016682520641442377757166, 14.40746688757498264634187925692, 15.47580004397535763731025518177, 16.03454446870892622541883554003, 17.05118468860295077528526639666, 18.13237086725818642391038489555, 18.935107468268763537688419778863, 19.60667244037059327838621014110, 20.38418608370430755108320178367, 21.45882938174666402536697810374, 21.7277697989182453352371506018, 22.601757715330085022445887990056

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