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Properties

Label 1-1792-1792.1493-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.0878 - 0.996i$
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  + (−0.352 − 0.935i)3-s + (−0.729 − 0.683i)5-s + (−0.751 + 0.659i)9-s + (−0.582 + 0.812i)11-s + (0.290 − 0.956i)13-s + (−0.382 + 0.923i)15-s + (0.991 + 0.130i)17-s + (−0.849 − 0.528i)19-s + (0.997 + 0.0654i)23-s + (0.0654 + 0.997i)25-s + (0.881 + 0.471i)27-s + (0.0980 + 0.995i)29-s + (0.965 − 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.999 + 0.0327i)37-s + ⋯
L(s)  = 1  + (−0.352 − 0.935i)3-s + (−0.729 − 0.683i)5-s + (−0.751 + 0.659i)9-s + (−0.582 + 0.812i)11-s + (0.290 − 0.956i)13-s + (−0.382 + 0.923i)15-s + (0.991 + 0.130i)17-s + (−0.849 − 0.528i)19-s + (0.997 + 0.0654i)23-s + (0.0654 + 0.997i)25-s + (0.881 + 0.471i)27-s + (0.0980 + 0.995i)29-s + (0.965 − 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.999 + 0.0327i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0878 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0878 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.0878 - 0.996i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.0878 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7706204796 - 0.7056812371i\)
\(L(\frac12)\) \(\approx\) \(0.7706204796 - 0.7056812371i\)
\(L(1)\) \(\approx\) \(0.7622049837 - 0.3223892435i\)
\(L(1)\) \(\approx\) \(0.7622049837 - 0.3223892435i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.352 - 0.935i)T \)
5 \( 1 + (-0.729 - 0.683i)T \)
11 \( 1 + (-0.582 + 0.812i)T \)
13 \( 1 + (0.290 - 0.956i)T \)
17 \( 1 + (0.991 + 0.130i)T \)
19 \( 1 + (-0.849 - 0.528i)T \)
23 \( 1 + (0.997 + 0.0654i)T \)
29 \( 1 + (0.0980 + 0.995i)T \)
31 \( 1 + (0.965 - 0.258i)T \)
37 \( 1 + (-0.999 + 0.0327i)T \)
41 \( 1 + (0.831 + 0.555i)T \)
43 \( 1 + (0.634 + 0.773i)T \)
47 \( 1 + (-0.793 - 0.608i)T \)
53 \( 1 + (0.812 + 0.582i)T \)
59 \( 1 + (-0.973 - 0.227i)T \)
61 \( 1 + (0.162 - 0.986i)T \)
67 \( 1 + (0.935 - 0.352i)T \)
71 \( 1 + (0.195 - 0.980i)T \)
73 \( 1 + (0.321 - 0.946i)T \)
79 \( 1 + (0.130 + 0.991i)T \)
83 \( 1 + (0.471 + 0.881i)T \)
89 \( 1 + (-0.442 + 0.896i)T \)
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

−20.753674667581152661989747921873, −19.2743813859197964496470066640, −19.151558626369707469126491143749, −18.28394836456459214516279358466, −17.20863409671381278070795485866, −16.64258506434588192620977273693, −15.81799940351149642314244519778, −15.45654683335515978603625628111, −14.40258107451055668568514705171, −14.09394456347070172484875005076, −12.83190389871850412526934022787, −11.81132065409331883068062213071, −11.41915912766249039514364800107, −10.53376175828139847819751937771, −10.14629039904109634466801390898, −8.9685043831117859091015225637, −8.36860254279422948063146085738, −7.41472055230969940174397791900, −6.43766978884980070543129886685, −5.77332023904154879656405809128, −4.76644418081466751000799186858, −3.9513404623137740237941424309, −3.312376490229105477100384112171, −2.43539857316782926736359019964, −0.760480738588049571454378342072, 0.60911212819861741374848167032, 1.43880832405387234512389652260, 2.60098464339235234827636744030, 3.47974206468649514536993679377, 4.81051864929834898106385702522, 5.192063385278488230280454738085, 6.258388309563474340944872011966, 7.15415773603287545148317478966, 7.86856933231458774476646623925, 8.342330655033800557829983898940, 9.32135837269392284412567508824, 10.49372181506734677456430229147, 11.07543119429388644120122211607, 12.034656356452618942839518841804, 12.73744075752114933619250307451, 12.901481140661557934324613275386, 13.95574731869331075535776808077, 15.01931903408465893030689613170, 15.5319063441583373958938736678, 16.5269391045969513915219688753, 17.12379478599995237948919205121, 17.84790553063830158600540883786, 18.54390426410609011363918284260, 19.39352479497241414816002855924, 19.81592343238659038333503116603

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