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Properties

Label 1-1292-1292.455-r0-0-0
Degree $1$
Conductor $1292$
Sign $-0.615 - 0.788i$
Analytic cond. $6.00002$
Root an. cond. $6.00002$
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  + i·3-s + i·5-s + i·7-s − 9-s + i·11-s − 13-s − 15-s − 21-s + i·23-s − 25-s − i·27-s − i·29-s + i·31-s − 33-s − 35-s + ⋯
L(s)  = 1  + i·3-s + i·5-s + i·7-s − 9-s + i·11-s − 13-s − 15-s − 21-s + i·23-s − 25-s − i·27-s − i·29-s + i·31-s − 33-s − 35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1292 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1292 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1292\)    =    \(2^{2} \cdot 17 \cdot 19\)
Sign: $-0.615 - 0.788i$
Analytic conductor: \(6.00002\)
Root analytic conductor: \(6.00002\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1292} (455, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1292,\ (0:\ ),\ -0.615 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3801839979 + 0.7791799478i\)
\(L(\frac12)\) \(\approx\) \(-0.3801839979 + 0.7791799478i\)
\(L(1)\) \(\approx\) \(0.6011486323 + 0.6358083685i\)
\(L(1)\) \(\approx\) \(0.6011486323 + 0.6358083685i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
19 \( 1 \)
good3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + iT \)
11 \( 1 \)
13 \( 1 + iT \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 - T \)
47 \( 1 \)
53 \( 1 - T \)
59 \( 1 \)
61 \( 1 \)
67 \( 1 \)
71 \( 1 \)
73 \( 1 \)
79 \( 1 - T \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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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

−20.36358863967990002742098066421, −19.70991035009143693355385109413, −19.14411150593834546236705922146, −18.22438392314261032905739484049, −17.32421027229433208966905994969, −16.75387679449231428850403209039, −16.33227839313950205254749442246, −14.97854026274458168833570203831, −14.05643102722492817398474239611, −13.55437926316600784672842341649, −12.77258087665016378564408495690, −12.171868881426703197297136243113, −11.29681792903431856857820430335, −10.43806660352928903838476430007, −9.35690610822032671418135941207, −8.516038622003312261973302499437, −7.8498831933153485096636066711, −7.07939179834862151726316744768, −6.185216040136170646007631025881, −5.27534069173133640812116832563, −4.406222117498450618761778815165, −3.307754645353743303145128338618, −2.21022193399528126090030034317, −1.12302701466948077372533290812, −0.35841235030641965954569037768, 2.05520236777279395292509053523, 2.695970064941119001447724739000, 3.59928674017093605666250814049, 4.62527473393651297401376200621, 5.39929665510980787727155004596, 6.244600825898857185239641965480, 7.25717841894812227107198556518, 8.090378112466474687956917770729, 9.28736349089275977291300865704, 9.7177886095252399353116740098, 10.454146356104219417804141360117, 11.423897485692696202286137853224, 11.94872140999461546720489031283, 12.92110076119677451560440501769, 14.28603811114002105694060427723, 14.59195162277901874515124945813, 15.51377266495578944167721393602, 15.69860095619600537041461832604, 17.02068405748409877877514786452, 17.667280321183054057483084845638, 18.322908783320985859312702354173, 19.45047972484676491577906693685, 19.77740524587823861401895661641, 21.00175066999358003143743446692, 21.57729527627492987380686245098

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