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Properties

Label 1-967-967.448-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.946 - 0.321i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (−0.0941 − 0.995i)2-s + (0.995 − 0.0974i)3-s + (−0.982 + 0.187i)4-s + (−0.779 + 0.626i)5-s + (−0.190 − 0.981i)6-s + (−0.228 + 0.973i)7-s + (0.279 + 0.960i)8-s + (0.981 − 0.193i)9-s + (0.696 + 0.717i)10-s + (−0.799 + 0.600i)11-s + (−0.959 + 0.282i)12-s + (−0.919 + 0.392i)13-s + (0.990 + 0.136i)14-s + (−0.715 + 0.699i)15-s + (0.929 − 0.368i)16-s + (−0.750 + 0.660i)17-s + ⋯
L(s)  = 1  + (−0.0941 − 0.995i)2-s + (0.995 − 0.0974i)3-s + (−0.982 + 0.187i)4-s + (−0.779 + 0.626i)5-s + (−0.190 − 0.981i)6-s + (−0.228 + 0.973i)7-s + (0.279 + 0.960i)8-s + (0.981 − 0.193i)9-s + (0.696 + 0.717i)10-s + (−0.799 + 0.600i)11-s + (−0.959 + 0.282i)12-s + (−0.919 + 0.392i)13-s + (0.990 + 0.136i)14-s + (−0.715 + 0.699i)15-s + (0.929 − 0.368i)16-s + (−0.750 + 0.660i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.946 - 0.321i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (448, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.946 - 0.321i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02031332846 + 0.1229061016i\)
\(L(\frac12)\) \(\approx\) \(0.02031332846 + 0.1229061016i\)
\(L(1)\) \(\approx\) \(0.8228823998 - 0.06276868759i\)
\(L(1)\) \(\approx\) \(0.8228823998 - 0.06276868759i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.0941 - 0.995i)T \)
3 \( 1 + (0.995 - 0.0974i)T \)
5 \( 1 + (-0.779 + 0.626i)T \)
7 \( 1 + (-0.228 + 0.973i)T \)
11 \( 1 + (-0.799 + 0.600i)T \)
13 \( 1 + (-0.919 + 0.392i)T \)
17 \( 1 + (-0.750 + 0.660i)T \)
19 \( 1 + (-0.663 + 0.748i)T \)
23 \( 1 + (-0.999 + 0.0390i)T \)
29 \( 1 + (0.0292 + 0.999i)T \)
31 \( 1 + (0.771 + 0.636i)T \)
37 \( 1 + (-0.177 - 0.984i)T \)
41 \( 1 + (0.334 + 0.942i)T \)
43 \( 1 + (-0.643 - 0.765i)T \)
47 \( 1 + (-0.840 - 0.541i)T \)
53 \( 1 + (0.113 + 0.993i)T \)
59 \( 1 + (0.943 + 0.331i)T \)
61 \( 1 + (0.983 + 0.181i)T \)
67 \( 1 + (-0.165 - 0.986i)T \)
71 \( 1 + (0.909 - 0.416i)T \)
73 \( 1 + (0.113 - 0.993i)T \)
79 \( 1 + (-0.847 + 0.530i)T \)
83 \( 1 + (0.254 - 0.967i)T \)
89 \( 1 + (0.936 + 0.350i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−20.95770617869778340311061991128, −20.21323193735590048346175035997, −19.43729987497816889945993118932, −19.00220119017853873812741143347, −17.80420991413404276647704045769, −17.04079727199727990263947783070, −16.06863674798345994746826496883, −15.71633129851374467450594585334, −14.9199789243780891750739830373, −14.00205967303131927960996971555, −13.2338734254859195252376878008, −12.883059821592734407262582928650, −11.44912835393567125860049988627, −10.173570564990258968325393835650, −9.649286462889723126056081405013, −8.50590801671736095339993716822, −8.06184457517634251321205963148, −7.362320627838963207071572349105, −6.54946741996978090273932966332, −5.05937185445451343459071170870, −4.44029178583664919506403108268, −3.67316480432339685677482260234, −2.502247821494817995785778205515, −0.75599665581129392891105624724, −0.03147841417441608111011969525, 1.91423679029412710136834043189, 2.362602693489953023423462478189, 3.2654628212435486503730362139, 4.11862193771001473314029383319, 4.982648376850975112375285500016, 6.484589531537680831395988721396, 7.58166566578979844002832502654, 8.29875567202114950649652322376, 8.973000431694160995001313670658, 10.0282987000232382549550021438, 10.486350149089052022662305176462, 11.72900149069378886605942165114, 12.42525470260373293105308969325, 12.90641719672955294190751713317, 14.09799130399468640593621230959, 14.77917653558384767765022607891, 15.350424481695345358956740206464, 16.31695078084276162086027870112, 17.81561161477860875133028718402, 18.32343953114082380539618844058, 19.07185978576612824520400055574, 19.63646662810488868277222141156, 20.15053083752274099952219044011, 21.29255557408136376864302198994, 21.697763667128468172229063159826

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