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Properties

Label 1-4235-4235.1159-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.756 - 0.654i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.879 − 0.475i)2-s + (0.104 − 0.994i)3-s + (0.548 + 0.836i)4-s + (−0.564 + 0.825i)6-s + (−0.0855 − 0.996i)8-s + (−0.978 − 0.207i)9-s + (0.888 − 0.458i)12-s + (0.254 − 0.967i)13-s + (−0.398 + 0.917i)16-s + (−0.345 − 0.938i)17-s + (0.761 + 0.647i)18-s + (0.272 + 0.962i)19-s + (0.995 + 0.0950i)23-s + (−0.999 − 0.0190i)24-s + (−0.683 + 0.730i)26-s + (−0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (−0.879 − 0.475i)2-s + (0.104 − 0.994i)3-s + (0.548 + 0.836i)4-s + (−0.564 + 0.825i)6-s + (−0.0855 − 0.996i)8-s + (−0.978 − 0.207i)9-s + (0.888 − 0.458i)12-s + (0.254 − 0.967i)13-s + (−0.398 + 0.917i)16-s + (−0.345 − 0.938i)17-s + (0.761 + 0.647i)18-s + (0.272 + 0.962i)19-s + (0.995 + 0.0950i)23-s + (−0.999 − 0.0190i)24-s + (−0.683 + 0.730i)26-s + (−0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.756 - 0.654i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.756 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3806285111 - 1.022162830i\)
\(L(\frac12)\) \(\approx\) \(0.3806285111 - 1.022162830i\)
\(L(1)\) \(\approx\) \(0.6213573515 - 0.4400456889i\)
\(L(1)\) \(\approx\) \(0.6213573515 - 0.4400456889i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.879 - 0.475i)T \)
3 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (0.254 - 0.967i)T \)
17 \( 1 + (-0.345 - 0.938i)T \)
19 \( 1 + (0.272 + 0.962i)T \)
23 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (-0.466 - 0.884i)T \)
31 \( 1 + (0.161 - 0.986i)T \)
37 \( 1 + (0.625 + 0.780i)T \)
41 \( 1 + (0.610 + 0.791i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.761 - 0.647i)T \)
53 \( 1 + (0.398 + 0.917i)T \)
59 \( 1 + (-0.991 - 0.132i)T \)
61 \( 1 + (-0.851 - 0.524i)T \)
67 \( 1 + (-0.235 - 0.971i)T \)
71 \( 1 + (0.897 + 0.441i)T \)
73 \( 1 + (-0.00951 - 0.999i)T \)
79 \( 1 + (0.797 - 0.603i)T \)
83 \( 1 + (0.736 + 0.676i)T \)
89 \( 1 + (0.928 + 0.371i)T \)
97 \( 1 + (-0.516 + 0.856i)T \)
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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

−18.582369358232666228145848253071, −17.78257222383133427416639303624, −17.151918440283492155949391703648, −16.66162030743177426191950252739, −15.8735212353459369856460961742, −15.54155253343493072380958135967, −14.6800249630563409201491872590, −14.26803977583896800478518720342, −13.42306213321795768617083897852, −12.34549116601667925709295885724, −11.40733684139706359682731030983, −10.828101248765088331755746659085, −10.49055179098155253904861422733, −9.38050124406172680571008670704, −9.0344057068248630107997752911, −8.58615668230507496806567388435, −7.5332326395796552196511101642, −6.865199149833042262146315285381, −6.07047847627256067737585828751, −5.311057663524597666226873855624, −4.60234135095194733836082162594, −3.768827751198890332778426915998, −2.76518772190503313068330928275, −1.97530878565814178217927124862, −0.879591062588020514197708768855, 0.53981209710601504706739022265, 1.162694178607855938262246112794, 2.13939177388143010964550655198, 2.83334400268469643122186048067, 3.43495742865771957200166566309, 4.57414194740757969217994716975, 5.77185704626496419630095665576, 6.31229284374952835851019967218, 7.24471221216838038620232990705, 7.83493737304235655240384850514, 8.19371636101189404252552118874, 9.28449162964752843406211870681, 9.58795266149556922380711906202, 10.77282464935378516005839006437, 11.163270820336516096733064421795, 12.027430104231454037345690902164, 12.45371733254574700781298375100, 13.384047046557756792545524400080, 13.610109762994059252568089293194, 14.88460652260020070978121018506, 15.38245237376559687849972815770, 16.41401533717133880659086155882, 16.94812004000067331156268154204, 17.61803154659950598591708456565, 18.29626817395213836556769852482

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