Properties

Degree 1
Conductor $ 5 \cdot 7 $
Sign $-0.525 + 0.850i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + i·2-s + i·3-s − 4-s − 6-s i·8-s − 9-s + 11-s i·12-s + i·13-s + 16-s i·17-s i·18-s + 19-s + i·22-s i·23-s + 24-s + ⋯
L(s,χ)  = 1  + i·2-s + i·3-s − 4-s − 6-s i·8-s − 9-s + 11-s i·12-s + i·13-s + 16-s i·17-s i·18-s + 19-s + i·22-s i·23-s + 24-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.525 + 0.850i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.525 + 0.850i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(35\)    =    \(5 \cdot 7\)
\( \varepsilon \)  =  $-0.525 + 0.850i$
motivic weight  =  \(0\)
character  :  $\chi_{35} (27, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 35,\ (0:\ ),\ -0.525 + 0.850i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3495409364 + 0.6269381942i$
$L(\frac12,\chi)$  $\approx$  $0.3495409364 + 0.6269381942i$
$L(\chi,1)$  $\approx$  0.6429206321 + 0.6213619628i
$L(1,\chi)$  $\approx$  0.6429206321 + 0.6213619628i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−35.59951234669536118631027473796, −34.96594794508327945164366480788, −32.89805787003820697711851844002, −31.58000120611266879858115315066, −30.409235903601587578397042002710, −29.75398098699249022140680697189, −28.51995644866412899747302258081, −27.39878541779762140002021896241, −25.788321567429246981409529946750, −24.37413874887608792004563219858, −23.041337673810567924880400625073, −21.98639976815761551022228892823, −20.24072717160410528073128794936, −19.445296194174099175945502728693, −18.15074251271976619161510547886, −17.15671754869575145751683826601, −14.6330808216027073292113375627, −13.33076328509740566163682877234, −12.262193970198583798876640901354, −11.053478916741687481603872758834, −9.2554330327191394876459260761, −7.74177751885047395888392449373, −5.69811009151214174983702061306, −3.41948173395721615043337197672, −1.527642942546248751768372213376, 3.82563061124272229542433742487, 5.20395516969527222914434958646, 6.862885020203486068328139135464, 8.78447499880181833223265346231, 9.748062668948478834491865400376, 11.68350960227318976088753411709, 13.860554686079885706438368123375, 14.817159354460680897899718186, 16.196602777548024283312635350635, 16.9594712992140305343597984306, 18.57036781802642855189416732928, 20.31396637660893572313074337635, 21.9182490332402576380203841768, 22.690978629151542201098086262251, 24.22286625632355759831920275751, 25.48973812595570355198839599415, 26.65109622148002811558886546996, 27.4602651989291724224164615184, 28.74320900269888855242547135730, 30.84274987882135682248030961034, 31.93405194990359413445943294049, 32.9838331568376425808078271417, 33.76268559702125704427475143284, 34.96311678411762509296875963934, 36.219896895435381643065199808892

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