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Properties

Label 1-107-107.14-r0-0-0
Degree $1$
Conductor $107$
Sign $0.835 + 0.549i$
Analytic cond. $0.496905$
Root an. cond. $0.496905$
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  + (−0.861 + 0.508i)2-s + (0.937 + 0.348i)3-s + (0.482 − 0.875i)4-s + (−0.998 − 0.0592i)5-s + (−0.984 + 0.176i)6-s + (0.582 − 0.812i)7-s + (0.0296 + 0.999i)8-s + (0.757 + 0.652i)9-s + (0.889 − 0.456i)10-s + (0.674 + 0.737i)11-s + (0.757 − 0.652i)12-s + (0.375 − 0.926i)13-s + (−0.0887 + 0.996i)14-s + (−0.915 − 0.403i)15-s + (−0.533 − 0.845i)16-s + (0.972 + 0.234i)17-s + ⋯
L(s)  = 1  + (−0.861 + 0.508i)2-s + (0.937 + 0.348i)3-s + (0.482 − 0.875i)4-s + (−0.998 − 0.0592i)5-s + (−0.984 + 0.176i)6-s + (0.582 − 0.812i)7-s + (0.0296 + 0.999i)8-s + (0.757 + 0.652i)9-s + (0.889 − 0.456i)10-s + (0.674 + 0.737i)11-s + (0.757 − 0.652i)12-s + (0.375 − 0.926i)13-s + (−0.0887 + 0.996i)14-s + (−0.915 − 0.403i)15-s + (−0.533 − 0.845i)16-s + (0.972 + 0.234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.835 + 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8633419657 + 0.2582726637i\)
\(L(\frac12)\) \(\approx\) \(0.8633419657 + 0.2582726637i\)
\(L(1)\) \(\approx\) \(0.8911449009 + 0.2091979000i\)
\(L(1)\) \(\approx\) \(0.8911449009 + 0.2091979000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.861 + 0.508i)T \)
3 \( 1 + (0.937 + 0.348i)T \)
5 \( 1 + (-0.998 - 0.0592i)T \)
7 \( 1 + (0.582 - 0.812i)T \)
11 \( 1 + (0.674 + 0.737i)T \)
13 \( 1 + (0.375 - 0.926i)T \)
17 \( 1 + (0.972 + 0.234i)T \)
19 \( 1 + (-0.717 + 0.696i)T \)
23 \( 1 + (-0.0887 - 0.996i)T \)
29 \( 1 + (-0.205 + 0.978i)T \)
31 \( 1 + (0.263 + 0.964i)T \)
37 \( 1 + (0.147 - 0.989i)T \)
41 \( 1 + (-0.794 - 0.606i)T \)
43 \( 1 + (-0.998 + 0.0592i)T \)
47 \( 1 + (-0.794 + 0.606i)T \)
53 \( 1 + (-0.861 - 0.508i)T \)
59 \( 1 + (-0.205 - 0.978i)T \)
61 \( 1 + (0.582 + 0.812i)T \)
67 \( 1 + (0.0296 - 0.999i)T \)
71 \( 1 + (0.937 - 0.348i)T \)
73 \( 1 + (-0.956 + 0.292i)T \)
79 \( 1 + (-0.430 - 0.902i)T \)
83 \( 1 + (-0.320 + 0.947i)T \)
89 \( 1 + (-0.984 - 0.176i)T \)
97 \( 1 + (0.889 - 0.456i)T \)
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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

−29.75244209099276382557553994925, −28.221221849259555183571614305282, −27.466850665701619367740085270861, −26.60109928437692560856488196762, −25.60284500056244765546265649312, −24.615989795583764448840881802557, −23.67944289901507339887578344164, −21.802368964181929492360292376368, −21.0167385943986320299297956125, −19.88887859429835470570855321044, −18.93809886042448155833758547748, −18.62720370164514090971603059617, −17.018574281971032708756631148728, −15.74428368089458659657579275610, −14.80296876675989326721555730615, −13.36326953390912457815757516955, −11.86220167313765460061371900481, −11.458823734911026795779843012434, −9.586219429521271832084240592066, −8.57820540305017769549138664530, −7.93402000249816549546707062104, −6.62713918422086665468229760163, −4.08904953439529524681511515370, −2.95794098381527022066430297614, −1.48653297622429575440924865355, 1.494010531897157102693682615336, 3.52860533953620700231919518441, 4.82590221923622177983864427221, 6.92720366766697960688047223848, 7.92739971598855134099998521986, 8.56148176002512797237844235766, 10.07105666353398455758654191723, 10.860795634419125071934150212387, 12.50698433983187814436781663763, 14.393446123516684078107154843207, 14.8129921389250142674006175290, 16.00273725971296493048142846336, 16.92679115578867154822424456959, 18.281307831600135194784857034053, 19.40782022585684661386113745104, 20.16197235333156040425761150773, 20.83150536709116489940720285969, 22.84187898176313826554047014760, 23.73714271211976824205913885901, 24.895279533131967804376264772310, 25.681844929638692692681489757901, 26.82170987698121377670556892785, 27.43418978125446807537487207345, 28.061284758539666789932906197702, 29.97881765003021532830120595527

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