Properties

Label 1-1792-1792.1237-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.585 - 0.810i$
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  + (0.352 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.585 - 0.810i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.585 - 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8113498651 + 1.587429582i\)
\(L(\frac12)\) \(\approx\) \(-0.8113498651 + 1.587429582i\)
\(L(1)\) \(\approx\) \(0.8680099864 + 0.7870512719i\)
\(L(1)\) \(\approx\) \(0.8680099864 + 0.7870512719i\)

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

−19.54923014326106568488022524624, −18.78799316935106360152006797373, −18.01089442347759754380120588972, −17.47353646377810071137931041394, −16.83425526414446298714675777924, −15.78816986994376645258303652062, −15.10617043939006653596641166240, −14.07517852727832692890933481986, −13.33833400892568897878219176129, −13.14396469155839616098781540101, −12.268040037633826092906169849518, −11.38003212266029596992871163715, −10.53292802324718911138266822435, −9.52807924713181150592724145573, −8.76963393943836164988063713895, −8.23726858988179019645596437116, −7.30883516680254364557502453414, −6.523394732182437656455742037519, −5.520063598758065943669152468856, −5.16434638570819439986537380240, −3.697345970466186772662678832574, −2.711389975725756804925260382240, −2.087907101820202022321361949000, −0.90847452896186744842358592730, −0.31884684675250717769288362180, 1.62571881550771726451669046258, 2.42689449243879252417042023433, 3.18139422343379632008122157975, 4.16825074504878047573585373048, 5.04568014168074255145499646819, 5.645085344264574468016011869459, 6.92884755780571407564359231392, 7.317566598652955953418703273047, 8.67943393807187807376782388691, 9.270087158055095590722224144815, 9.92823778140295094366384870037, 10.665730073311603533916978177467, 11.21791433554883645159175125954, 12.28264403897579110779999753882, 13.29233762583975233510069408819, 14.00444663397026913311947107236, 14.58111877883407550009201284844, 15.275557184394736312742776481, 15.98500784512592962759638513130, 16.796447129144453456061578158532, 17.5415773529856485210456540237, 18.262487181669657971587157640976, 19.02342189124014191240317147468, 19.924944874269947247775847778663, 20.61988692173541983994302481327

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