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
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: | $\chi_{967} (632, \cdot )$ |
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
$p$ | $F_p(T)$ | |
---|---|---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (-0.158 + 0.987i)T \) |
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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Imaginary part of the first few zeros on the critical line
−21.28048264828260592529362971344, −20.57008090613907701076125525213, −19.709306627675278457550255119349, −18.924198249665829589867660533718, −17.95734212117060797643033029924, −17.26054683415611469263707729552, −16.88711987896920280281490561943, −15.84819623714269811429647778361, −14.87447363209491593229462801942, −13.64303497271196110702128622227, −13.14108551896118655190017174884, −11.86189533567008834186595114265, −11.61409123706505807375808126915, −11.15148897388358148852975002801, −9.84680139991906731453582546011, −9.126598418795179817808566038513, −8.54275567027383234649637453615, −7.34322915407353407525822063896, −6.03307852298864489575798623233, −5.08109194838614545747921241639, −4.42579793325544629092142396078, −3.820939456741222541983572511, −2.06012131268609228660745095971, −1.2988175970434888788499149978, −0.36537312300157777248124015403, 0.88780462694405094604204162514, 2.011327188394713534222071306824, 3.77532374978610251535740289822, 4.59533932748721922997963047864, 5.50779686422099960850868293039, 6.347773928074365801853914833164, 7.0728628128432686679519970619, 7.67954000145173587034903601767, 8.60982740821499867705706451937, 9.94658694654134775326694450018, 10.552513699260749058836610053862, 11.339760002481401026155968056679, 12.29995005162897420042394064642, 13.21732610558454258397308060142, 14.307368206478251683902700365987, 14.71831858566007265908656122893, 15.59951949803438001887853855692, 16.45565368076237044851088418400, 17.33771872549530765537936235274, 17.94431411113613446015693030034, 18.21942574337732093040870717031, 19.238247188013510831965098126701, 20.192013825862543271034701919479, 21.469580373069060140964995288923, 22.401949057726633128928619631187