Properties

Label 1-967-967.222-r0-0-0
Degree $1$
Conductor $967$
Sign $0.841 - 0.539i$
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  + (−0.184 + 0.982i)2-s + (−0.0292 + 0.999i)3-s + (−0.932 − 0.362i)4-s + (0.494 − 0.869i)5-s + (−0.977 − 0.212i)6-s + (0.0876 − 0.996i)7-s + (0.527 − 0.849i)8-s + (−0.998 − 0.0585i)9-s + (0.763 + 0.646i)10-s + (0.874 + 0.485i)11-s + (0.389 − 0.921i)12-s + (0.874 − 0.485i)13-s + (0.962 + 0.269i)14-s + (0.854 + 0.519i)15-s + (0.737 + 0.675i)16-s + (−0.995 + 0.0974i)17-s + ⋯
L(s)  = 1  + (−0.184 + 0.982i)2-s + (−0.0292 + 0.999i)3-s + (−0.932 − 0.362i)4-s + (0.494 − 0.869i)5-s + (−0.977 − 0.212i)6-s + (0.0876 − 0.996i)7-s + (0.527 − 0.849i)8-s + (−0.998 − 0.0585i)9-s + (0.763 + 0.646i)10-s + (0.874 + 0.485i)11-s + (0.389 − 0.921i)12-s + (0.874 − 0.485i)13-s + (0.962 + 0.269i)14-s + (0.854 + 0.519i)15-s + (0.737 + 0.675i)16-s + (−0.995 + 0.0974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.841 - 0.539i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.841 - 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8387979798 - 0.2459356223i\)
\(L(\frac12)\) \(\approx\) \(0.8387979798 - 0.2459356223i\)
\(L(1)\) \(\approx\) \(0.8127993388 + 0.2631076073i\)
\(L(1)\) \(\approx\) \(0.8127993388 + 0.2631076073i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.184 + 0.982i)T \)
3 \( 1 + (-0.0292 + 0.999i)T \)
5 \( 1 + (0.494 - 0.869i)T \)
7 \( 1 + (0.0876 - 0.996i)T \)
11 \( 1 + (0.874 + 0.485i)T \)
13 \( 1 + (0.874 - 0.485i)T \)
17 \( 1 + (-0.995 + 0.0974i)T \)
19 \( 1 + (-0.967 - 0.250i)T \)
23 \( 1 + (-0.297 - 0.954i)T \)
29 \( 1 + (0.165 - 0.986i)T \)
31 \( 1 + (-0.107 + 0.994i)T \)
37 \( 1 + (-0.668 - 0.744i)T \)
41 \( 1 + (-0.775 + 0.631i)T \)
43 \( 1 + (-0.544 - 0.838i)T \)
47 \( 1 + (-0.696 + 0.717i)T \)
53 \( 1 + (0.682 + 0.730i)T \)
59 \( 1 + (-0.864 - 0.502i)T \)
61 \( 1 + (-0.775 + 0.631i)T \)
67 \( 1 + (-0.107 - 0.994i)T \)
71 \( 1 + (-0.184 - 0.982i)T \)
73 \( 1 + (0.682 - 0.730i)T \)
79 \( 1 + (-0.145 - 0.989i)T \)
83 \( 1 + (0.972 - 0.232i)T \)
89 \( 1 + (-0.107 - 0.994i)T \)
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

−21.82088154070055517712679727707, −21.16339538703865312190947702939, −20.0267612074309940018448492809, −19.28035839300895839765753727297, −18.69480922692761884799143688861, −18.19409675975296604736418599030, −17.50077333689423952927279244798, −16.68577942476601813534335694022, −15.212710094674154277549687503737, −14.36497651712164666862247668387, −13.633357411552107483591703819089, −13.07683250611587426021673702432, −11.997220448976972078417701801883, −11.42482827442293389849982501517, −10.87868355208314263817034881115, −9.62625905913153350160541015560, −8.784930819228228223708440996194, −8.288641009624466003574750561655, −6.883607679728600418601276332262, −6.23212451790580938126123917292, −5.346522854486681418806096197000, −3.84096273501400798912181698867, −2.9806673820109179508630395156, −2.002897487208499656913363032636, −1.52450484346346958851479789627, 0.40054946825194427675726691102, 1.727760114757171116507412150440, 3.62411169398898120464775525095, 4.44835405420004760513103290633, 4.81743582282343800931962002548, 6.10971345783142242394574086412, 6.59237999706568444852643230332, 7.989004567961332298889262688630, 8.75877656329734263089917447064, 9.2579222770293136093920566036, 10.31118251346364903971403832399, 10.73599433825202643308697356817, 12.18177758450959021480358115571, 13.27897187843717680665434813358, 13.825064614640618072030887659539, 14.68993234007965490256408562806, 15.48660835021372802616993159681, 16.228888409576686062349418976765, 16.94745883513985796766499819678, 17.364973826069070676704666028686, 18.067476085076793301585468956389, 19.6127174918101719679065147553, 20.06804936269636301237167669020, 20.928829288376423618371036498271, 21.731369489283472607092296714

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