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Properties

Label 1-629-629.322-r0-0-0
Degree $1$
Conductor $629$
Sign $-0.621 - 0.783i$
Analytic cond. $2.92106$
Root an. cond. $2.92106$
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  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s − 6-s + (0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s − 10-s − 11-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s − 14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s − 6-s + (0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s − 10-s − 11-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s − 14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(629\)    =    \(17 \cdot 37\)
Sign: $-0.621 - 0.783i$
Analytic conductor: \(2.92106\)
Root analytic conductor: \(2.92106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{629} (322, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 629,\ (0:\ ),\ -0.621 - 0.783i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3449197763 + 0.7139512546i\)
\(L(\frac12)\) \(\approx\) \(-0.3449197763 + 0.7139512546i\)
\(L(1)\) \(\approx\) \(0.4677847841 + 0.6849017559i\)
\(L(1)\) \(\approx\) \(0.4677847841 + 0.6849017559i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 - T \)
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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

−22.2348828681745906204140465109, −21.109453997693638938097270221896, −20.67886595903404660798110742978, −20.07958437866693610026546183715, −19.15890016073370519467956196273, −18.414076003684806335902262802861, −17.588670045584677664835983430614, −16.98232973631281634157973035835, −16.13053354852534562766951667570, −14.39296828887523773316611565037, −13.870372762633280715286077468173, −12.95250139308007258600242017758, −12.482800337648786805213671908271, −11.48897648761321806577022978977, −10.447597903123340284956315382593, −9.55924868419618869675774727766, −8.719109074233301143508625632301, −7.86099379645841199659280455605, −7.29989606443714216826953400824, −5.79142942994410185117878228165, −4.54036947108564370890890586178, −3.64355185235802831669922888693, −2.097650738096730210890050547254, −1.78435557150533939659389165229, −0.40245759363355038734234942856, 2.10360621021304705162076912264, 2.81471677092666559410045323110, 4.3390835331479144929626290243, 5.45078219397071170096556133534, 5.82655091645966625928256957096, 7.361225537796162053727176605056, 7.98921303829621483782682220689, 8.98732285955095636338830031219, 9.71083670521877679512790868516, 10.58399683828652211174420876448, 11.11325930082579653967062956003, 12.8576031349363344044622342436, 13.844425927224846487107833801602, 14.652798431293517011247964134353, 15.24054063406258616211905972546, 15.70525451619285489060964133210, 16.83516752922950059002419401267, 17.80208552159498955260628663513, 18.30194108154241547990854721495, 19.204141822230348483416992713873, 20.12904000975524555077341505869, 21.14464242076694422966355940383, 22.031689588314766690541580213677, 22.43137492469960109283293012657, 23.61902034536632680026290188412

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