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Properties

Label 1-51-51.8-r1-0-0
Degree $1$
Conductor $51$
Sign $0.758 - 0.651i$
Analytic cond. $5.48071$
Root an. cond. $5.48071$
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 − 4-s + (−0.707 − 0.707i)5-s + (0.707 − 0.707i)7-s − i·8-s + (0.707 − 0.707i)10-s + (0.707 − 0.707i)11-s − 13-s + (0.707 + 0.707i)14-s + 16-s − i·19-s + (0.707 + 0.707i)20-s + (0.707 + 0.707i)22-s + (0.707 − 0.707i)23-s + i·25-s − i·26-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (−0.707 − 0.707i)5-s + (0.707 − 0.707i)7-s − i·8-s + (0.707 − 0.707i)10-s + (0.707 − 0.707i)11-s − 13-s + (0.707 + 0.707i)14-s + 16-s − i·19-s + (0.707 + 0.707i)20-s + (0.707 + 0.707i)22-s + (0.707 − 0.707i)23-s + i·25-s − i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.758 - 0.651i$
Analytic conductor: \(5.48071\)
Root analytic conductor: \(5.48071\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 51,\ (1:\ ),\ 0.758 - 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9652921016 - 0.3575308607i\)
\(L(\frac12)\) \(\approx\) \(0.9652921016 - 0.3575308607i\)
\(L(1)\) \(\approx\) \(0.8772849573 + 0.06676618322i\)
\(L(1)\) \(\approx\) \(0.8772849573 + 0.06676618322i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + iT \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 - T \)
19 \( 1 - iT \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + iT \)
47 \( 1 + T \)
53 \( 1 + iT \)
59 \( 1 - iT \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 + T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + iT \)
89 \( 1 + T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−33.441024021787737128000406945795, −31.76333560010512351232944609698, −31.03628060081439389726050062646, −30.14551807796394889731844442463, −28.96329538845831248011940638627, −27.448817304907034220010796835426, −27.24107020314851591095771552351, −25.52742391042694189390730261215, −23.92351708643886824578360725138, −22.64203551146934000583132483857, −21.85709821896756605777071252643, −20.48563516481680998189202430212, −19.34938417978731451351698433805, −18.408584536665463655784011142042, −17.221437451009870591643102934080, −15.08300864326840368423467042426, −14.31995872839081239751098762927, −12.359995732940766689835838546760, −11.65577010615909971514756492015, −10.34105930459807525340476589854, −8.89059161696449549295724013672, −7.38308499457201030117071254895, −5.1143299805046261660436760694, −3.58963077346673468424265339321, −1.95216987762580434627851044638, 0.62596751968808266921754691008, 4.02630910764597911312567761433, 5.12563508344194356407176726687, 7.009090631543033714669496214476, 8.12007488484863153823019010150, 9.31472709836302817874466035866, 11.27469222439326877347311200438, 12.83750348471725525893483588292, 14.171681420707755321779594076100, 15.27281807804285149715564011380, 16.68629501479845818372629817987, 17.27455030711882944005258476169, 18.97682742597191163796622058775, 20.163413472538948460457356335329, 21.75571861150222612401279992909, 23.08622198950769692804491828297, 24.21280603154102275529237713458, 24.669339358269383555736168922342, 26.49634769653298107036424943804, 27.17644648003566737495129522843, 28.190842512105779204938973122, 29.97787965725781766813092912118, 31.22597598907043564107884077281, 32.22733185414175701442402606195, 33.103109169275282523179486736982

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