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Properties

Label 1-41-41.29-r1-0-0
Degree $1$
Conductor $41$
Sign $-0.996 - 0.0817i$
Analytic cond. $4.40606$
Root an. cond. $4.40606$
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  + (−0.951 − 0.309i)2-s + (−0.707 − 0.707i)3-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (0.453 + 0.891i)6-s + (0.453 − 0.891i)7-s + (−0.587 − 0.809i)8-s + i·9-s + (−0.809 + 0.587i)10-s + (−0.987 − 0.156i)11-s + (−0.156 − 0.987i)12-s + (−0.891 + 0.453i)13-s + (−0.707 + 0.707i)14-s + (−0.987 + 0.156i)15-s + (0.309 + 0.951i)16-s + (0.156 − 0.987i)17-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.707 − 0.707i)3-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (0.453 + 0.891i)6-s + (0.453 − 0.891i)7-s + (−0.587 − 0.809i)8-s + i·9-s + (−0.809 + 0.587i)10-s + (−0.987 − 0.156i)11-s + (−0.156 − 0.987i)12-s + (−0.891 + 0.453i)13-s + (−0.707 + 0.707i)14-s + (−0.987 + 0.156i)15-s + (0.309 + 0.951i)16-s + (0.156 − 0.987i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $-0.996 - 0.0817i$
Analytic conductor: \(4.40606\)
Root analytic conductor: \(4.40606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (1:\ ),\ -0.996 - 0.0817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02189373088 - 0.5346498201i\)
\(L(\frac12)\) \(\approx\) \(0.02189373088 - 0.5346498201i\)
\(L(1)\) \(\approx\) \(0.4297786542 - 0.3463863244i\)
\(L(1)\) \(\approx\) \(0.4297786542 - 0.3463863244i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + (0.453 - 0.891i)T \)
11 \( 1 + (-0.987 - 0.156i)T \)
13 \( 1 + (-0.891 + 0.453i)T \)
17 \( 1 + (0.156 - 0.987i)T \)
19 \( 1 + (-0.891 - 0.453i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.156 + 0.987i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (0.453 + 0.891i)T \)
53 \( 1 + (-0.156 - 0.987i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.951 - 0.309i)T \)
67 \( 1 + (0.987 - 0.156i)T \)
71 \( 1 + (0.987 + 0.156i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (0.453 - 0.891i)T \)
97 \( 1 + (-0.987 + 0.156i)T \)
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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

−34.546978161698735161295302864833, −34.41620255612556305048202506955, −33.36609028470677344036579769341, −32.05733956444826926642905907380, −30.11859503089220953306562589733, −28.90635672895350266638744749924, −28.09623855486877897855118288210, −26.89339204095101650144871112935, −26.00266402332160196862383885799, −24.71360378548312418759092813098, −23.24771625188047442851100483818, −21.78777150726642654278836780254, −20.867234351083986959419244970716, −18.92752960035971797156307468423, −17.906014105523921481089733085966, −17.04285912742853771434533347311, −15.42328492919489071798789916710, −14.77489653230356944744903800998, −12.15153010444679290977033698638, −10.63287270294607065640455096157, −9.99536487554827543452106031783, −8.292826136034057803731004116544, −6.427379553722475068638470976334, −5.310917012728344034681837960303, −2.416037374089381033131630599089, 0.46976774120970056758449707372, 2.0389195648619784410001961874, 5.05315019567408128064878194230, 6.934983987725556939516575951816, 8.11854568902230660386407017369, 9.85587801878180304501912522529, 11.12160307633187294725403450967, 12.44210520553825014089723071204, 13.64107924468574698096208565519, 16.15657030942219820122699113254, 17.16328473623324839723520581870, 17.875436043665925471843836293293, 19.27978795786291281624129989908, 20.54482166388406079217892803875, 21.67799997304079203658729214851, 23.64754526941719597401581895608, 24.48539560966881737968298997275, 25.75433531578781128538102281519, 27.19716710255724369231982022257, 28.30564198729144139608756527076, 29.345176666217347782450445782254, 29.86620377344137075094035674567, 31.52496460133282447721533974184, 33.51958542818512577873556730689, 34.0882667806105283575367781102

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