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Properties

Label 1-1259-1259.1076-r1-0-0
Degree $1$
Conductor $1259$
Sign $0.705 - 0.708i$
Analytic cond. $135.298$
Root an. cond. $135.298$
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  + (0.186 + 0.982i)2-s + (−0.476 + 0.879i)3-s + (−0.930 + 0.365i)4-s + (−0.0972 − 0.995i)5-s + (−0.952 − 0.304i)6-s + (0.498 + 0.866i)7-s + (−0.532 − 0.846i)8-s + (−0.545 − 0.838i)9-s + (0.959 − 0.280i)10-s + (−0.928 + 0.370i)11-s + (0.122 − 0.992i)12-s + (−0.998 − 0.0598i)13-s + (−0.758 + 0.651i)14-s + (0.921 + 0.389i)15-s + (0.732 − 0.681i)16-s + (0.868 + 0.496i)17-s + ⋯
L(s)  = 1  + (0.186 + 0.982i)2-s + (−0.476 + 0.879i)3-s + (−0.930 + 0.365i)4-s + (−0.0972 − 0.995i)5-s + (−0.952 − 0.304i)6-s + (0.498 + 0.866i)7-s + (−0.532 − 0.846i)8-s + (−0.545 − 0.838i)9-s + (0.959 − 0.280i)10-s + (−0.928 + 0.370i)11-s + (0.122 − 0.992i)12-s + (−0.998 − 0.0598i)13-s + (−0.758 + 0.651i)14-s + (0.921 + 0.389i)15-s + (0.732 − 0.681i)16-s + (0.868 + 0.496i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $0.705 - 0.708i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1076, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ 0.705 - 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4491003206 + 0.1866810541i\)
\(L(\frac12)\) \(\approx\) \(-0.4491003206 + 0.1866810541i\)
\(L(1)\) \(\approx\) \(0.4310999009 + 0.5823899852i\)
\(L(1)\) \(\approx\) \(0.4310999009 + 0.5823899852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (0.186 + 0.982i)T \)
3 \( 1 + (-0.476 + 0.879i)T \)
5 \( 1 + (-0.0972 - 0.995i)T \)
7 \( 1 + (0.498 + 0.866i)T \)
11 \( 1 + (-0.928 + 0.370i)T \)
13 \( 1 + (-0.998 - 0.0598i)T \)
17 \( 1 + (0.868 + 0.496i)T \)
19 \( 1 + (0.0274 + 0.999i)T \)
23 \( 1 + (-0.707 + 0.706i)T \)
29 \( 1 + (0.292 + 0.956i)T \)
31 \( 1 + (-0.590 + 0.806i)T \)
37 \( 1 + (0.326 + 0.945i)T \)
41 \( 1 + (-0.873 + 0.487i)T \)
43 \( 1 + (0.913 + 0.407i)T \)
47 \( 1 + (0.979 - 0.203i)T \)
53 \( 1 + (-0.992 - 0.124i)T \)
59 \( 1 + (0.432 + 0.901i)T \)
61 \( 1 + (0.515 + 0.856i)T \)
67 \( 1 + (0.630 + 0.776i)T \)
71 \( 1 + (-0.671 + 0.740i)T \)
73 \( 1 + (-0.283 + 0.959i)T \)
79 \( 1 + (0.606 - 0.794i)T \)
83 \( 1 + (0.445 - 0.895i)T \)
89 \( 1 + (-0.718 - 0.695i)T \)
97 \( 1 + (-0.244 - 0.969i)T \)
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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

−20.15560878486111086270477069472, −19.26108262021540192750685927591, −18.83425745835560628209831723416, −18.03505110297204238409598011915, −17.52534669772426153280711452980, −16.699069326885754222725144660318, −15.41308400195630149414807717494, −14.25559554586213075928221598456, −14.024253273862172580493584818179, −13.17603135633092560083990167741, −12.31009124945596578121032032795, −11.57322839065196769896561643601, −10.86754738152571944918830076820, −10.41197260122201049423433988437, −9.47085740326228871766033205609, −7.942799523005653718971438264512, −7.65377734523623301253718942505, −6.61404857852190342917439897862, −5.561393003165241687983573207296, −4.81175055999674090101750445556, −3.72048742711884568604896445510, −2.554152160342766995997458079154, −2.17915962967725450048564987680, −0.67080948627096549259445336900, −0.14720843037909365669424177788, 1.38262298051595758692693018903, 2.98495552355253020830506378630, 4.08742275669899912314914615922, 4.89761775600676320865981734192, 5.43374352988628214749423156476, 5.9001430905011106900013482400, 7.363132806228801656009861284292, 8.17302302447791455608923013505, 8.77797942596927045843678865079, 9.747676539643052247082153388287, 10.227218811026707550474253441338, 11.73345070611453276967990360139, 12.3231537269874634856852052872, 12.851893674244037055947119353583, 14.213849274834216229055057162155, 14.797378955252511135720337598465, 15.53443919331975315383066648649, 16.14724840966350543422850561441, 16.75352010853130626757149293927, 17.53649787625460051526873306104, 18.09415578109709496843495320411, 19.09109336427447263727613927205, 20.31613233649952866489020490298, 20.9927863070158347529772733978, 21.72614817638800884956810817757

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