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Properties

Label 1-1792-1792.803-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.624 + 0.780i$
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  + (−0.582 + 0.812i)3-s + (−0.528 + 0.849i)5-s + (−0.321 − 0.946i)9-s + (−0.935 + 0.352i)11-s + (−0.881 − 0.471i)13-s + (−0.382 − 0.923i)15-s + (−0.608 − 0.793i)17-s + (−0.729 − 0.683i)19-s + (−0.896 − 0.442i)23-s + (−0.442 − 0.896i)25-s + (0.956 + 0.290i)27-s + (0.634 + 0.773i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (0.973 − 0.227i)37-s + ⋯
L(s)  = 1  + (−0.582 + 0.812i)3-s + (−0.528 + 0.849i)5-s + (−0.321 − 0.946i)9-s + (−0.935 + 0.352i)11-s + (−0.881 − 0.471i)13-s + (−0.382 − 0.923i)15-s + (−0.608 − 0.793i)17-s + (−0.729 − 0.683i)19-s + (−0.896 − 0.442i)23-s + (−0.442 − 0.896i)25-s + (0.956 + 0.290i)27-s + (0.634 + 0.773i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (0.973 − 0.227i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.624 + 0.780i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.624 + 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5496350578 + 0.2641168703i\)
\(L(\frac12)\) \(\approx\) \(0.5496350578 + 0.2641168703i\)
\(L(1)\) \(\approx\) \(0.5991425428 + 0.1968963002i\)
\(L(1)\) \(\approx\) \(0.5991425428 + 0.1968963002i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.582 + 0.812i)T \)
5 \( 1 + (-0.528 + 0.849i)T \)
11 \( 1 + (-0.935 + 0.352i)T \)
13 \( 1 + (-0.881 - 0.471i)T \)
17 \( 1 + (-0.608 - 0.793i)T \)
19 \( 1 + (-0.729 - 0.683i)T \)
23 \( 1 + (-0.896 - 0.442i)T \)
29 \( 1 + (0.634 + 0.773i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (0.973 - 0.227i)T \)
41 \( 1 + (0.555 + 0.831i)T \)
43 \( 1 + (0.995 - 0.0980i)T \)
47 \( 1 + (0.130 + 0.991i)T \)
53 \( 1 + (0.352 + 0.935i)T \)
59 \( 1 + (-0.0327 + 0.999i)T \)
61 \( 1 + (-0.910 + 0.412i)T \)
67 \( 1 + (-0.812 - 0.582i)T \)
71 \( 1 + (0.980 - 0.195i)T \)
73 \( 1 + (0.751 + 0.659i)T \)
79 \( 1 + (0.793 + 0.608i)T \)
83 \( 1 + (-0.290 - 0.956i)T \)
89 \( 1 + (0.0654 - 0.997i)T \)
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

−19.718760055989924250106001229597, −19.5002834019716580554786362850, −18.68573417381839126210927557497, −17.80677652196515692744897529988, −17.19654600473521564141396726000, −16.47526523150822221752659359532, −15.894789170549329648268119050066, −14.9961892469421780026311103438, −13.94147157161662579513088887004, −13.27137202068740275306839119569, −12.424554192246084619608701682643, −12.192967199474134527958074164920, −11.18389675317220831183724543210, −10.52226004023428766032620656998, −9.51103132930828616116971588176, −8.33352940061380445845867850771, −8.05471652362457092229539557695, −7.15777545940551711483977273465, −6.21371456262918176255331005755, −5.469232284588161225956745228823, −4.67494745278743320864737700123, −3.85990978472196897022121582934, −2.43070377564325377237189058404, −1.74310482418363440035552883788, −0.496546809415831391531409632346, 0.47888539547195647419400311897, 2.52970230908962134655241146537, 2.83086535428585558778640420268, 4.23461213223716345386684269686, 4.58471038150906724947345671252, 5.64056196588346972458348542310, 6.453955781104185655597325176965, 7.32610337467239078828438974730, 8.01063748610187893749775385667, 9.190703723447664905041169637916, 9.89065896247289621851453488314, 10.74943119299884148674607252874, 11.00984640861935445888814004490, 12.04609675390468208485907040620, 12.62643899441778062713047585242, 13.73976787010752201545231772213, 14.73336195693693840041138120730, 15.15046822645882990222370283287, 15.84976928648896259009935492481, 16.45152278451271376385753675302, 17.518636839796791809640734893020, 18.00726701038639548797959255294, 18.64952087295461590588498632624, 19.86756794329944826534086886404, 20.12495054812643836055455850405

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