Properties

Label 1-35-35.13-r0-0-0
Degree $1$
Conductor $35$
Sign $-0.525 - 0.850i$
Analytic cond. $0.162539$
Root an. cond. $0.162539$
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·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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(0.162539\)
Root analytic conductor: \(0.162539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 35,\ (0:\ ),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3495409364 - 0.6269381942i\)
\(L(\frac12)\) \(\approx\) \(0.3495409364 - 0.6269381942i\)
\(L(1)\) \(\approx\) \(0.6429206321 - 0.6213619628i\)
\(L(1)\) \(\approx\) \(0.6429206321 - 0.6213619628i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 - T \)
19 \( 1 \)
23 \( 1 + iT \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 + T \)
41 \( 1 + iT \)
43 \( 1 - iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 + iT \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 - iT \)
89 \( 1 + iT \)
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

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

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