Properties

Label 1-967-967.46-r1-0-0
Degree $1$
Conductor $967$
Sign $0.891 + 0.453i$
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  + (−0.322 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.891 + 0.453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.655946405 + 0.3973622440i\)
\(L(\frac12)\) \(\approx\) \(1.655946405 + 0.3973622440i\)
\(L(1)\) \(\approx\) \(0.9624236871 + 0.3420926082i\)
\(L(1)\) \(\approx\) \(0.9624236871 + 0.3420926082i\)

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

−21.20409393588690255836619885306, −20.634735669514319763624102064166, −20.0050980038177072564678657961, −19.453304320378426720710281627798, −18.597522941627251728828863300542, −17.63211192513881920481539958939, −16.69156818741066053238540445444, −16.17462440153263042881119491644, −15.2248660759976701288908311741, −13.72506304968778165569923323270, −13.63652760917522185583033171855, −12.80846483822650532817534701641, −11.91598509881803121610079083658, −10.80892056013844911185607818869, −9.82052298364173776651263341367, −9.539018863405546644668540628459, −8.85901699421601246692073246079, −7.77745079305089094410413716681, −7.060087391183018873731081592095, −5.06482376488252267255033347028, −4.80956001117156951111811432261, −3.588693078472186494923395068927, −2.801147244220416118998474888348, −1.93486830223205641519595345600, −0.71126247048873440062412803448, 0.50578537885214415820375412354, 2.06933776890931606701911462197, 2.90556796451311075449356194608, 3.81879479385650489809860726602, 5.41255844911199288888265146403, 6.0766338806176658244345375791, 6.91359866221235277534965929872, 7.67372964497034489972788608577, 8.34854134615327440098810639317, 9.420841401533090999990522829710, 9.959060313527024314001128534999, 10.77045694907375646535457474757, 12.40435130166992481354584059897, 13.13127345812995127470952603585, 13.69376027393351964964076094485, 14.82695759038741246777809938932, 15.01945579019572495287376828723, 15.88719280527836311694417971703, 16.91250470201568490443846811390, 17.90494676023223927306106635726, 18.464193717992308523872838160436, 19.05408833812631455956170488290, 19.630231789157902766324305305663, 20.79291783483625053131220339482, 21.787415921162604921446553369800

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