Properties

Label 1-4011-4011.824-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.376 - 0.926i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
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  + (0.997 − 0.0770i)2-s + (0.988 − 0.153i)4-s + (−0.998 − 0.0550i)5-s + (0.973 − 0.229i)8-s + (−0.999 + 0.0220i)10-s + (−0.975 + 0.218i)11-s + (0.148 + 0.988i)13-s + (0.952 − 0.303i)16-s + (−0.556 + 0.831i)17-s + (−0.999 − 0.0220i)19-s + (−0.995 + 0.0990i)20-s + (−0.956 + 0.293i)22-s + (−0.329 + 0.944i)23-s + (0.993 + 0.110i)25-s + (0.224 + 0.974i)26-s + ⋯
L(s)  = 1  + (0.997 − 0.0770i)2-s + (0.988 − 0.153i)4-s + (−0.998 − 0.0550i)5-s + (0.973 − 0.229i)8-s + (−0.999 + 0.0220i)10-s + (−0.975 + 0.218i)11-s + (0.148 + 0.988i)13-s + (0.952 − 0.303i)16-s + (−0.556 + 0.831i)17-s + (−0.999 − 0.0220i)19-s + (−0.995 + 0.0990i)20-s + (−0.956 + 0.293i)22-s + (−0.329 + 0.944i)23-s + (0.993 + 0.110i)25-s + (0.224 + 0.974i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.376 - 0.926i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.376 - 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6923681977 - 1.028500353i\)
\(L(\frac12)\) \(\approx\) \(0.6923681977 - 1.028500353i\)
\(L(1)\) \(\approx\) \(1.329607828 - 0.1254237768i\)
\(L(1)\) \(\approx\) \(1.329607828 - 0.1254237768i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.997 - 0.0770i)T \)
5 \( 1 + (-0.998 - 0.0550i)T \)
11 \( 1 + (-0.975 + 0.218i)T \)
13 \( 1 + (0.148 + 0.988i)T \)
17 \( 1 + (-0.556 + 0.831i)T \)
19 \( 1 + (-0.999 - 0.0220i)T \)
23 \( 1 + (-0.329 + 0.944i)T \)
29 \( 1 + (-0.746 - 0.665i)T \)
31 \( 1 + (0.191 - 0.981i)T \)
37 \( 1 + (-0.191 - 0.981i)T \)
41 \( 1 + (-0.879 - 0.475i)T \)
43 \( 1 + (-0.340 - 0.940i)T \)
47 \( 1 + (0.815 + 0.578i)T \)
53 \( 1 + (0.660 - 0.750i)T \)
59 \( 1 + (-0.421 - 0.906i)T \)
61 \( 1 + (-0.938 - 0.345i)T \)
67 \( 1 + (0.884 - 0.466i)T \)
71 \( 1 + (0.724 - 0.689i)T \)
73 \( 1 + (0.999 + 0.0110i)T \)
79 \( 1 + (-0.949 - 0.314i)T \)
83 \( 1 + (0.652 - 0.757i)T \)
89 \( 1 + (-0.899 - 0.436i)T \)
97 \( 1 + (0.0165 - 0.999i)T \)
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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

−18.59070071752327948015891986881, −18.259792916200124840590103981785, −17.052680115871658686769276335516, −16.41472788601602667939665079847, −15.77964761040053103315025465765, −15.23654872193615420657161999495, −14.803294488077497854756190184674, −13.814173440645318070421885541232, −13.21994762330875942665929006120, −12.51393960226210304446801858428, −12.075333034637026804588260784365, −11.04179785737606719720271778966, −10.793805980854157570492033262998, −10.008291572484474103257371944551, −8.503331336452836384674290462720, −8.26848736299921801085270996230, −7.313019815028314712122128098267, −6.797728470389993066236508071630, −5.892152020370265686143234915317, −5.03520519180867495689293710383, −4.57023787048541793435151812473, −3.664724115251642137741116799808, −2.94499406204571001402421011427, −2.397371482508215997764647184860, −1.059396359707485896379849097079, 0.23958755797422489746366366132, 1.852981216682713495797048354060, 2.23966170863636951757990998745, 3.475110309137653847316098781773, 3.97962660697278643927951922827, 4.57254344448585127877210523903, 5.42038015411790916090908475020, 6.22372533663248040036665142760, 6.97395426741679031385110765561, 7.67208785960961523377393284501, 8.27304934614730094668363953945, 9.23117695878132488756217373744, 10.27391495578684708148856504411, 10.956825491233927471867338945750, 11.42698107960961167016578693353, 12.20115492698241039280613884535, 12.78839145708920266418303669922, 13.427934209932686488739684873379, 14.11534119517680101520694483775, 15.10718998351384958484293938513, 15.34919552074738472518528827905, 15.93893880092168766089403351617, 16.80107422830226657132565730149, 17.31141371605783671973368954545, 18.64543684527615291919566084987

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