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Properties

Label 1-1792-1792.1173-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.999 - 0.0388i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
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  + (0.162 − 0.986i)3-s + (−0.973 − 0.227i)5-s + (−0.946 − 0.321i)9-s + (0.910 − 0.412i)11-s + (−0.956 − 0.290i)13-s + (−0.382 + 0.923i)15-s + (−0.608 + 0.793i)17-s + (0.999 + 0.0327i)19-s + (0.442 + 0.896i)23-s + (0.896 + 0.442i)25-s + (−0.471 + 0.881i)27-s + (0.995 − 0.0980i)29-s + (−0.258 + 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.849 + 0.528i)37-s + ⋯
L(s)  = 1  + (0.162 − 0.986i)3-s + (−0.973 − 0.227i)5-s + (−0.946 − 0.321i)9-s + (0.910 − 0.412i)11-s + (−0.956 − 0.290i)13-s + (−0.382 + 0.923i)15-s + (−0.608 + 0.793i)17-s + (0.999 + 0.0327i)19-s + (0.442 + 0.896i)23-s + (0.896 + 0.442i)25-s + (−0.471 + 0.881i)27-s + (0.995 − 0.0980i)29-s + (−0.258 + 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.849 + 0.528i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.999 - 0.0388i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.999 - 0.0388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063179704 + 0.02064503013i\)
\(L(\frac12)\) \(\approx\) \(1.063179704 + 0.02064503013i\)
\(L(1)\) \(\approx\) \(0.8635434097 - 0.2286850673i\)
\(L(1)\) \(\approx\) \(0.8635434097 - 0.2286850673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.162 - 0.986i)T \)
5 \( 1 + (-0.973 - 0.227i)T \)
11 \( 1 + (0.910 - 0.412i)T \)
13 \( 1 + (-0.956 - 0.290i)T \)
17 \( 1 + (-0.608 + 0.793i)T \)
19 \( 1 + (0.999 + 0.0327i)T \)
23 \( 1 + (0.442 + 0.896i)T \)
29 \( 1 + (0.995 - 0.0980i)T \)
31 \( 1 + (-0.258 + 0.965i)T \)
37 \( 1 + (-0.849 + 0.528i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (-0.773 + 0.634i)T \)
47 \( 1 + (-0.130 + 0.991i)T \)
53 \( 1 + (-0.412 - 0.910i)T \)
59 \( 1 + (0.729 + 0.683i)T \)
61 \( 1 + (0.352 + 0.935i)T \)
67 \( 1 + (0.986 + 0.162i)T \)
71 \( 1 + (-0.195 + 0.980i)T \)
73 \( 1 + (-0.659 - 0.751i)T \)
79 \( 1 + (0.793 - 0.608i)T \)
83 \( 1 + (0.881 - 0.471i)T \)
89 \( 1 + (-0.997 + 0.0654i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−20.15705473951267662754959049386, −19.71825623588599491374185611247, −18.89928312266084699089340605833, −17.99160299547067966426355212233, −16.9947132907729593350322834517, −16.52717993354866487255953336389, −15.63507957196111817313613111531, −15.1750356008684835882822021315, −14.38393069942615754595228159519, −13.88185871290745597213893730599, −12.53199604868834441902281091176, −11.7445016275245522842826599837, −11.374598593772922297067268443431, −10.36131189088784666223541316609, −9.63987452372218468876047277750, −8.94744794815419941713858145630, −8.16156757710460594293846013660, −7.16198948051671958245702932574, −6.6039264297650957799546451190, −5.141177493160546185690697953098, −4.66883797552861447258573114114, −3.84702737367461904391069011835, −3.0841504655504549076598434475, −2.16851365401487754659905192491, −0.47331461775288726228194541422, 0.901088544736682396434231710714, 1.70323724352329191532171570566, 3.01432734155087391190136620682, 3.55022160426739569774187240485, 4.719757345206805544813330039486, 5.566581869417191979631344079, 6.72086260911521597104669064825, 7.12091394042104054625705178317, 8.068427828571819835097347088955, 8.59878713126443531139536901185, 9.42294782891712536398434427471, 10.57137426161649959867221631099, 11.64573070920540445693721437134, 11.86135389140185667275080907341, 12.68721747721069472989405993997, 13.437850447779638175602787465073, 14.29196871828629800674149426165, 14.91344622854067449756125376327, 15.744122499487142497826599888601, 16.60434048725881340567521620586, 17.43335585409099141154319503632, 17.877215635992925148345965278452, 19.11600881920105981035590227504, 19.372175507362159730011885493110, 19.91480570178738922235853052834

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