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Properties

Label 1-1792-1792.179-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.280 + 0.959i$
Analytic cond. $192.577$
Root an. cond. $192.577$
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  + (0.729 + 0.683i)3-s + (0.412 − 0.910i)5-s + (0.0654 + 0.997i)9-s + (−0.528 + 0.849i)11-s + (0.0980 − 0.995i)13-s + (0.923 − 0.382i)15-s + (0.130 + 0.991i)17-s + (0.986 − 0.162i)19-s + (−0.751 − 0.659i)23-s + (−0.659 − 0.751i)25-s + (−0.634 + 0.773i)27-s + (−0.881 − 0.471i)29-s + (0.965 + 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.935 + 0.352i)37-s + ⋯
L(s)  = 1  + (0.729 + 0.683i)3-s + (0.412 − 0.910i)5-s + (0.0654 + 0.997i)9-s + (−0.528 + 0.849i)11-s + (0.0980 − 0.995i)13-s + (0.923 − 0.382i)15-s + (0.130 + 0.991i)17-s + (0.986 − 0.162i)19-s + (−0.751 − 0.659i)23-s + (−0.659 − 0.751i)25-s + (−0.634 + 0.773i)27-s + (−0.881 − 0.471i)29-s + (0.965 + 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.935 + 0.352i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.280 + 0.959i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.280 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.314793321 + 1.735237358i\)
\(L(\frac12)\) \(\approx\) \(2.314793321 + 1.735237358i\)
\(L(1)\) \(\approx\) \(1.407732642 + 0.2860422453i\)
\(L(1)\) \(\approx\) \(1.407732642 + 0.2860422453i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.729 + 0.683i)T \)
5 \( 1 + (0.412 - 0.910i)T \)
11 \( 1 + (-0.528 + 0.849i)T \)
13 \( 1 + (0.0980 - 0.995i)T \)
17 \( 1 + (0.130 + 0.991i)T \)
19 \( 1 + (0.986 - 0.162i)T \)
23 \( 1 + (-0.751 - 0.659i)T \)
29 \( 1 + (-0.881 - 0.471i)T \)
31 \( 1 + (0.965 + 0.258i)T \)
37 \( 1 + (-0.935 + 0.352i)T \)
41 \( 1 + (0.980 + 0.195i)T \)
43 \( 1 + (0.956 + 0.290i)T \)
47 \( 1 + (0.608 + 0.793i)T \)
53 \( 1 + (0.849 + 0.528i)T \)
59 \( 1 + (-0.812 + 0.582i)T \)
61 \( 1 + (-0.973 - 0.227i)T \)
67 \( 1 + (0.683 - 0.729i)T \)
71 \( 1 + (0.831 + 0.555i)T \)
73 \( 1 + (0.442 - 0.896i)T \)
79 \( 1 + (0.991 + 0.130i)T \)
83 \( 1 + (-0.773 + 0.634i)T \)
89 \( 1 + (-0.946 - 0.321i)T \)
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

−19.68172312319836133184770425464, −19.0353298233754628985590862574, −18.40999974437619802322584640417, −18.05358619892183432370002757507, −17.04749845453892272571797351051, −16.01905639973754800104892582860, −15.422972478726579658578105219865, −14.35732509982853871751823161969, −13.78622941632483834910629649382, −13.67437188443158953254878589231, −12.421203503215932538500654937998, −11.62499224448763578545026217796, −10.98522069010615776852267113809, −9.851785504316796090087575669502, −9.33552301763032453204282972727, −8.437719050845962855949195514882, −7.43681580192844065547455512979, −7.10393405683808923059691114568, −6.07522150877504245534433342262, −5.43492836425706010252454278970, −3.9490166081365513615505972448, −3.21313590042086152588627420025, −2.469697024176523363658345434580, −1.66633316001895784175679558327, −0.49851581204789350517733078432, 0.93526066353983280908422515921, 2.01097238393914146244445748249, 2.78691748995128630009469242847, 3.86895649084663981246734271767, 4.61091412354911584528420793490, 5.349307696415738005307274024773, 6.10272783223809766997985794185, 7.62101368960970502290480545388, 7.9967024785836392922501398784, 8.878454712385602889331041290281, 9.63027250012735072375677174888, 10.20856220212682546881870469616, 10.88762257606431997547730955849, 12.26699876465170731375095091911, 12.70607821029529144199377388630, 13.59089080306783403439661104157, 14.15327192719748067749604085817, 15.26224395362493769251515722740, 15.55782488282309267413332368242, 16.4027126738738648042009197895, 17.20760046719499279229616491936, 17.841032364358190750902697447059, 18.75548927356135820192970569156, 19.92187562311728839578854319094, 20.05581916775127974231511435791

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