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Properties

Label 1-967-967.687-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.920 - 0.390i$
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  + (−0.158 − 0.987i)2-s + (−0.962 + 0.269i)3-s + (−0.949 + 0.313i)4-s + (−0.113 − 0.993i)5-s + (0.419 + 0.907i)6-s + (0.974 + 0.225i)7-s + (0.460 + 0.887i)8-s + (0.854 − 0.519i)9-s + (−0.962 + 0.269i)10-s + (0.854 − 0.519i)11-s + (0.829 − 0.557i)12-s + (−0.0227 + 0.999i)13-s + (0.0682 − 0.997i)14-s + (0.377 + 0.926i)15-s + (0.803 − 0.595i)16-s + (−0.990 + 0.136i)17-s + ⋯
L(s)  = 1  + (−0.158 − 0.987i)2-s + (−0.962 + 0.269i)3-s + (−0.949 + 0.313i)4-s + (−0.113 − 0.993i)5-s + (0.419 + 0.907i)6-s + (0.974 + 0.225i)7-s + (0.460 + 0.887i)8-s + (0.854 − 0.519i)9-s + (−0.962 + 0.269i)10-s + (0.854 − 0.519i)11-s + (0.829 − 0.557i)12-s + (−0.0227 + 0.999i)13-s + (0.0682 − 0.997i)14-s + (0.377 + 0.926i)15-s + (0.803 − 0.595i)16-s + (−0.990 + 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.920 - 0.390i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2087396983 - 1.026847901i\)
\(L(\frac12)\) \(\approx\) \(0.2087396983 - 1.026847901i\)
\(L(1)\) \(\approx\) \(0.6190867992 - 0.4100940170i\)
\(L(1)\) \(\approx\) \(0.6190867992 - 0.4100940170i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.158 - 0.987i)T \)
3 \( 1 + (-0.962 + 0.269i)T \)
5 \( 1 + (-0.113 - 0.993i)T \)
7 \( 1 + (0.974 + 0.225i)T \)
11 \( 1 + (0.854 - 0.519i)T \)
13 \( 1 + (-0.0227 + 0.999i)T \)
17 \( 1 + (-0.990 + 0.136i)T \)
19 \( 1 + (0.247 - 0.968i)T \)
23 \( 1 + (-0.203 + 0.979i)T \)
29 \( 1 + (-0.854 - 0.519i)T \)
31 \( 1 + (0.538 - 0.842i)T \)
37 \( 1 + (-0.0227 + 0.999i)T \)
41 \( 1 + (0.576 - 0.816i)T \)
43 \( 1 + (-0.613 + 0.789i)T \)
47 \( 1 + (-0.377 - 0.926i)T \)
53 \( 1 + (0.746 + 0.665i)T \)
59 \( 1 + (0.746 - 0.665i)T \)
61 \( 1 + (0.995 + 0.0909i)T \)
67 \( 1 + (-0.460 + 0.887i)T \)
71 \( 1 + (-0.775 - 0.631i)T \)
73 \( 1 + (0.746 - 0.665i)T \)
79 \( 1 + (-0.746 + 0.665i)T \)
83 \( 1 + (-0.419 - 0.907i)T \)
89 \( 1 + (0.998 - 0.0455i)T \)
97 \( 1 + T \)
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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

−22.401949057726633128928619631187, −21.469580373069060140964995288923, −20.192013825862543271034701919479, −19.238247188013510831965098126701, −18.21942574337732093040870717031, −17.94431411113613446015693030034, −17.33771872549530765537936235274, −16.45565368076237044851088418400, −15.59951949803438001887853855692, −14.71831858566007265908656122893, −14.307368206478251683902700365987, −13.21732610558454258397308060142, −12.29995005162897420042394064642, −11.339760002481401026155968056679, −10.552513699260749058836610053862, −9.94658694654134775326694450018, −8.60982740821499867705706451937, −7.67954000145173587034903601767, −7.0728628128432686679519970619, −6.347773928074365801853914833164, −5.50779686422099960850868293039, −4.59533932748721922997963047864, −3.77532374978610251535740289822, −2.011327188394713534222071306824, −0.88780462694405094604204162514, 0.36537312300157777248124015403, 1.2988175970434888788499149978, 2.06012131268609228660745095971, 3.820939456741222541983572511, 4.42579793325544629092142396078, 5.08109194838614545747921241639, 6.03307852298864489575798623233, 7.34322915407353407525822063896, 8.54275567027383234649637453615, 9.126598418795179817808566038513, 9.84680139991906731453582546011, 11.15148897388358148852975002801, 11.61409123706505807375808126915, 11.86189533567008834186595114265, 13.14108551896118655190017174884, 13.64303497271196110702128622227, 14.87447363209491593229462801942, 15.84819623714269811429647778361, 16.88711987896920280281490561943, 17.26054683415611469263707729552, 17.95734212117060797643033029924, 18.924198249665829589867660533718, 19.709306627675278457550255119349, 20.57008090613907701076125525213, 21.28048264828260592529362971344

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