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Properties

Label 1-967-967.111-r0-0-0
Degree $1$
Conductor $967$
Sign $0.0643 + 0.997i$
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  + (−0.992 + 0.123i)2-s + (−0.999 + 0.0195i)3-s + (0.969 − 0.244i)4-s + (−0.941 + 0.337i)5-s + (0.989 − 0.142i)6-s + (0.448 − 0.893i)7-s + (−0.932 + 0.362i)8-s + (0.999 − 0.0390i)9-s + (0.892 − 0.451i)10-s + (−0.184 + 0.982i)11-s + (−0.964 + 0.263i)12-s + (0.943 + 0.331i)13-s + (−0.334 + 0.942i)14-s + (0.934 − 0.356i)15-s + (0.880 − 0.474i)16-s + (−0.442 − 0.896i)17-s + ⋯
L(s)  = 1  + (−0.992 + 0.123i)2-s + (−0.999 + 0.0195i)3-s + (0.969 − 0.244i)4-s + (−0.941 + 0.337i)5-s + (0.989 − 0.142i)6-s + (0.448 − 0.893i)7-s + (−0.932 + 0.362i)8-s + (0.999 − 0.0390i)9-s + (0.892 − 0.451i)10-s + (−0.184 + 0.982i)11-s + (−0.964 + 0.263i)12-s + (0.943 + 0.331i)13-s + (−0.334 + 0.942i)14-s + (0.934 − 0.356i)15-s + (0.880 − 0.474i)16-s + (−0.442 − 0.896i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0643 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0643 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.0643 + 0.997i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.0643 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3197174915 + 0.2997667578i\)
\(L(\frac12)\) \(\approx\) \(0.3197174915 + 0.2997667578i\)
\(L(1)\) \(\approx\) \(0.4567343024 + 0.08071477562i\)
\(L(1)\) \(\approx\) \(0.4567343024 + 0.08071477562i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.992 + 0.123i)T \)
3 \( 1 + (-0.999 + 0.0195i)T \)
5 \( 1 + (-0.941 + 0.337i)T \)
7 \( 1 + (0.448 - 0.893i)T \)
11 \( 1 + (-0.184 + 0.982i)T \)
13 \( 1 + (0.943 + 0.331i)T \)
17 \( 1 + (-0.442 - 0.896i)T \)
19 \( 1 + (-0.347 + 0.937i)T \)
23 \( 1 + (0.316 + 0.948i)T \)
29 \( 1 + (0.592 + 0.805i)T \)
31 \( 1 + (-0.997 + 0.0714i)T \)
37 \( 1 + (0.847 - 0.530i)T \)
41 \( 1 + (-0.0682 - 0.997i)T \)
43 \( 1 + (-0.927 - 0.374i)T \)
47 \( 1 + (0.861 + 0.508i)T \)
53 \( 1 + (-0.877 - 0.480i)T \)
59 \( 1 + (-0.171 + 0.985i)T \)
61 \( 1 + (0.898 + 0.439i)T \)
67 \( 1 + (0.560 - 0.828i)T \)
71 \( 1 + (0.389 + 0.921i)T \)
73 \( 1 + (-0.877 + 0.480i)T \)
79 \( 1 + (0.581 - 0.813i)T \)
83 \( 1 + (-0.359 - 0.933i)T \)
89 \( 1 + (0.436 + 0.899i)T \)
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

−21.57806391250290037133986785821, −20.73713292677849718028456717276, −19.83123989711546876362856195006, −18.858720599078822820644580728633, −18.5499192931502162734780812181, −17.67949888430744183089110237954, −16.85667291468942404363397269907, −16.16339387184540332291801778681, −15.51761113328800550277761556822, −14.96266382867201280110658860558, −13.12644048628263607217000251411, −12.53865313649717262836815690777, −11.507346801231453184933911884731, −11.2025252620615642356093822456, −10.55498235662038868544980291069, −9.233976468046902449161093150619, −8.38469407189403651814701825903, −8.010474489422833788733929894228, −6.65073309647166544226072126679, −6.07885022094354780068841450935, −5.02824709307516408486382987225, −3.92139660693091835421649963218, −2.7375192522062177566763593346, −1.41885207376739435323226947509, −0.412143911954727949068165904532, 0.94064492485376624064649463290, 1.89844005873690736813387921174, 3.53104701764852737606258915845, 4.40965948427812998209262065300, 5.47285608685151671931945336541, 6.68685975959899463075226410475, 7.20696324479953958513513051073, 7.80551440919499768199287792938, 8.94628607171683400778991447214, 10.02878810176003710115618856760, 10.73657562280651850772039025090, 11.26981908381019463785485306919, 11.94884815418372869965540050019, 12.86891195596786140286087971435, 14.20955302808127434537650950205, 15.148085563923675538546804467844, 15.92931662950884223699044077314, 16.4233441652152449684627862123, 17.301073722137632701722458784806, 18.052989390854489758397704555134, 18.50803824777941602690322897286, 19.4525482249164049871858620413, 20.36406740087570214992427887879, 20.820650192917291776030282736066, 21.97446836382048759422341208483

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