Properties

Label 1-967-967.21-r0-0-0
Degree $1$
Conductor $967$
Sign $0.936 - 0.351i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (−0.322 − 0.946i)2-s + (−0.864 − 0.502i)3-s + (−0.791 + 0.610i)4-s + (−0.395 + 0.918i)5-s + (−0.197 + 0.980i)6-s + (0.861 + 0.508i)7-s + (0.833 + 0.552i)8-s + (0.494 + 0.869i)9-s + (0.996 + 0.0779i)10-s + (−0.957 + 0.288i)11-s + (0.991 − 0.129i)12-s + (0.728 − 0.684i)13-s + (0.203 − 0.979i)14-s + (0.803 − 0.595i)15-s + (0.254 − 0.967i)16-s + (−0.184 − 0.982i)17-s + ⋯
L(s)  = 1  + (−0.322 − 0.946i)2-s + (−0.864 − 0.502i)3-s + (−0.791 + 0.610i)4-s + (−0.395 + 0.918i)5-s + (−0.197 + 0.980i)6-s + (0.861 + 0.508i)7-s + (0.833 + 0.552i)8-s + (0.494 + 0.869i)9-s + (0.996 + 0.0779i)10-s + (−0.957 + 0.288i)11-s + (0.991 − 0.129i)12-s + (0.728 − 0.684i)13-s + (0.203 − 0.979i)14-s + (0.803 − 0.595i)15-s + (0.254 − 0.967i)16-s + (−0.184 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.936 - 0.351i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.936 - 0.351i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7194965428 - 0.1306634733i\)
\(L(\frac12)\) \(\approx\) \(0.7194965428 - 0.1306634733i\)
\(L(1)\) \(\approx\) \(0.6176820308 - 0.1932034686i\)
\(L(1)\) \(\approx\) \(0.6176820308 - 0.1932034686i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.322 - 0.946i)T \)
3 \( 1 + (-0.864 - 0.502i)T \)
5 \( 1 + (-0.395 + 0.918i)T \)
7 \( 1 + (0.861 + 0.508i)T \)
11 \( 1 + (-0.957 + 0.288i)T \)
13 \( 1 + (0.728 - 0.684i)T \)
17 \( 1 + (-0.184 - 0.982i)T \)
19 \( 1 + (-0.783 + 0.620i)T \)
23 \( 1 + (-0.668 - 0.744i)T \)
29 \( 1 + (0.987 - 0.155i)T \)
31 \( 1 + (0.633 + 0.773i)T \)
37 \( 1 + (-0.0812 - 0.996i)T \)
41 \( 1 + (0.962 + 0.269i)T \)
43 \( 1 + (-0.993 - 0.110i)T \)
47 \( 1 + (0.966 + 0.257i)T \)
53 \( 1 + (0.995 + 0.0909i)T \)
59 \( 1 + (0.549 - 0.835i)T \)
61 \( 1 + (-0.715 + 0.699i)T \)
67 \( 1 + (0.353 + 0.935i)T \)
71 \( 1 + (0.981 - 0.193i)T \)
73 \( 1 + (0.995 - 0.0909i)T \)
79 \( 1 + (-0.857 + 0.514i)T \)
83 \( 1 + (0.999 - 0.0260i)T \)
89 \( 1 + (-0.986 - 0.161i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−21.607229747268359042640153294520, −21.293195217367984900438160295633, −20.26261344782818110985116751455, −19.29788251255885470366913795142, −18.323142172549644972670746818578, −17.584775745813771755493120553287, −16.966268563100543107294741576982, −16.39456778314356967652141566275, −15.51163388349666649876972197827, −15.176507892789201713306832661808, −13.80412580179874994661740555120, −13.218393460226560913792020655434, −12.1293455455562016096059095028, −11.12675272202900114187891750200, −10.52827410971464811718370521915, −9.56077489285937404405909673979, −8.48687550540711163430617660342, −8.10916791902599222045741567376, −6.94123036907536888561839969538, −6.0078140163231836035985976203, −5.22428086951771350229472034574, −4.422254706593846588562571336683, −3.95493284109188495279302158779, −1.62297745586231749470232669533, −0.6111193454497920072950234927, 0.8037250231813163361361801927, 2.159438605883344210076844465481, 2.697253546958025840364461984571, 4.09330794474435244284399534833, 4.97409904165517478457138652827, 5.90065633581336316984571323474, 7.0656327209314679520666224325, 7.98379636909243262933439925320, 8.43288189275147114825885149978, 10.02728883778057088173634328833, 10.66882725130485813413523685819, 11.122285828007971823224919588987, 12.04441416007894266436168743690, 12.50841325399815923550786757603, 13.58491568161669974899836563783, 14.32345471985608726136158393485, 15.51451291068355030040157140612, 16.23210311503156216581945609510, 17.49977770647445014858994915918, 18.128532133493845300005897757159, 18.32818798632777540665157425924, 19.117946119843938801736770760739, 20.087912287075715014398862815762, 21.059839717422260584377913309669, 21.58255786612778312167114742942

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