Properties

Label 1-287-287.237-r1-0-0
Degree $1$
Conductor $287$
Sign $0.943 - 0.331i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
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  − 2-s + i·3-s + 4-s + 5-s i·6-s − 8-s − 9-s − 10-s i·11-s + i·12-s + i·13-s + i·15-s + 16-s i·17-s + 18-s i·19-s + ⋯
L(s)  = 1  − 2-s + i·3-s + 4-s + 5-s i·6-s − 8-s − 9-s − 10-s i·11-s + i·12-s + i·13-s + i·15-s + 16-s i·17-s + 18-s i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.943 - 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.244374321 - 0.2119214532i\)
\(L(\frac12)\) \(\approx\) \(1.244374321 - 0.2119214532i\)
\(L(1)\) \(\approx\) \(0.8227200263 + 0.1044464862i\)
\(L(1)\) \(\approx\) \(0.8227200263 + 0.1044464862i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - iT \)
19 \( 1 \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 - iT \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 + iT \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 + T \)
71 \( 1 - iT \)
73 \( 1 + T \)
79 \( 1 \)
83 \( 1 + iT \)
89 \( 1 + T \)
97 \( 1 - iT \)
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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

−25.38010683091884008707982712740, −24.876551787690887051548309322644, −23.80249031190228844972799845607, −22.79627114455007876860879770137, −21.55707631452237126092065153854, −20.42827234458333910187694841794, −19.90131873850427421470742042557, −18.65715968772054974462181897042, −18.10899808043816626964410450192, −17.30356185938159205088407469707, −16.72268837795427718367580297149, −15.12401306906261956294001228270, −14.39257421434502557604803530932, −12.824967845853139949568798368986, −12.56426690691076525576967934898, −11.05055935713582216903740478404, −10.205836369858183463712413906101, −9.18376142791688544411802152163, −8.155750723401929983138159988255, −7.24273814014498593010515727608, −6.28072609462002170018562686201, −5.41591548530614723161241924222, −3.08512626982350952776889053748, −1.95106079803916044176297306184, −1.169805011957107214398396836289, 0.57188728472683410728204360526, 2.235592688447811375653258821792, 3.24780120474281393298424073653, 4.93125832791676840034835973289, 5.982930145617967908734131472446, 6.99490065485122487416323227854, 8.58637380499668026935134482886, 9.23880321926159444930431652827, 9.90223332086882740948076544757, 11.065793045804493239337152748887, 11.544239432617936544025468262181, 13.3368005615931728935872465840, 14.32676387510602036910052960202, 15.338760643275508828235800122310, 16.468734173286861425683319801259, 16.796687538961063476183743135, 17.87264379256810551766963151803, 18.79200739657776014029027422174, 19.82996405693112664225647880493, 20.84305987053159840438753988109, 21.43232488061507508471861407482, 22.1290875763838916947310074270, 23.60882728495903381678732303666, 24.7361570627664272543722965199, 25.4484503749328110177196680704

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