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Properties

Label 1-2784-2784.683-r0-0-0
Degree $1$
Conductor $2784$
Sign $0.880 - 0.473i$
Analytic cond. $12.9288$
Root an. cond. $12.9288$
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  + (−0.111 + 0.993i)5-s + (−0.974 + 0.222i)7-s + (−0.330 − 0.943i)11-s + (0.943 − 0.330i)13-s + 17-s + (0.532 + 0.846i)19-s + (−0.781 − 0.623i)23-s + (−0.974 − 0.222i)25-s + (−0.623 − 0.781i)31-s + (−0.111 − 0.993i)35-s + (0.943 + 0.330i)37-s − i·41-s + (−0.993 + 0.111i)43-s + (0.900 + 0.433i)47-s + (0.900 − 0.433i)49-s + ⋯
L(s)  = 1  + (−0.111 + 0.993i)5-s + (−0.974 + 0.222i)7-s + (−0.330 − 0.943i)11-s + (0.943 − 0.330i)13-s + 17-s + (0.532 + 0.846i)19-s + (−0.781 − 0.623i)23-s + (−0.974 − 0.222i)25-s + (−0.623 − 0.781i)31-s + (−0.111 − 0.993i)35-s + (0.943 + 0.330i)37-s − i·41-s + (−0.993 + 0.111i)43-s + (0.900 + 0.433i)47-s + (0.900 − 0.433i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2784 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2784 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2784\)    =    \(2^{5} \cdot 3 \cdot 29\)
Sign: $0.880 - 0.473i$
Analytic conductor: \(12.9288\)
Root analytic conductor: \(12.9288\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2784} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2784,\ (0:\ ),\ 0.880 - 0.473i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.183911641 - 0.2982088692i\)
\(L(\frac12)\) \(\approx\) \(1.183911641 - 0.2982088692i\)
\(L(1)\) \(\approx\) \(0.9403653223 + 0.05185577396i\)
\(L(1)\) \(\approx\) \(0.9403653223 + 0.05185577396i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
29 \( 1 \)
good5 \( 1 + (-0.111 + 0.993i)T \)
7 \( 1 + (-0.974 + 0.222i)T \)
11 \( 1 + (-0.330 - 0.943i)T \)
13 \( 1 + (0.943 - 0.330i)T \)
17 \( 1 + T \)
19 \( 1 + (0.532 + 0.846i)T \)
23 \( 1 + (-0.781 - 0.623i)T \)
31 \( 1 + (-0.623 - 0.781i)T \)
37 \( 1 + (0.943 + 0.330i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.993 + 0.111i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (0.111 - 0.993i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.846 - 0.532i)T \)
67 \( 1 + (0.330 - 0.943i)T \)
71 \( 1 + (0.433 - 0.900i)T \)
73 \( 1 + (-0.781 - 0.623i)T \)
79 \( 1 + (-0.900 + 0.433i)T \)
83 \( 1 + (-0.532 - 0.846i)T \)
89 \( 1 + (0.781 - 0.623i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−19.40411935546024460021026682219, −18.54827549873987282133128950646, −17.89492216159980354910958735531, −17.027603623958283663192899399150, −16.43629843447801574311351383888, −15.78960981497080167503776889998, −15.36427792192651148518721782600, −14.143385668760159568750524765664, −13.507166851547590595130227436718, −12.88436016169253780415948174288, −12.24820543460317847819326823523, −11.62739348128258077463364962691, −10.54544014128815155946543416863, −9.83864117392265976545659052748, −9.22773695217620495794842232071, −8.565841395752387467756747166375, −7.53638540428912594369695922251, −7.06195144677974311130609919200, −5.90470403764617643533521715411, −5.41190102441872317666912170907, −4.36451064749211807038574896712, −3.77137607305990054614826420763, −2.84234035389838583492986302564, −1.69702982767641029303504909885, −0.87032734787501222266407733227, 0.4820918624034503825262479879, 1.77478108083163219368287512159, 3.00645852920772662279149352841, 3.26034040188768971563629216131, 4.05524858116387192531237982479, 5.481355998422627404668803463787, 6.11524462152686592629862313715, 6.47867245664320957141145720015, 7.76351313686689484304856872371, 8.03714034095531878314121873395, 9.18320577950899511243054245525, 9.97487356729483139912983473104, 10.48672432195967227960740513229, 11.29018771166587142214058382946, 11.96826179402944956344151234962, 12.85715718414632680845488939941, 13.54435778766726323722482577734, 14.21049486829393969936888447556, 14.91979426491816068744918424529, 15.75135103771882703954729523446, 16.29561308834640938327607744864, 16.84395185367428166363657338895, 18.13857813570210489535575517299, 18.6203882972277318368649383494, 18.81514972483279382054647651887

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