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Properties

Label 1-967-967.139-r0-0-0
Degree $1$
Conductor $967$
Sign $0.703 + 0.710i$
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  + (−0.741 − 0.670i)2-s + (−0.977 + 0.212i)3-s + (0.100 + 0.994i)4-s + (−0.922 − 0.386i)5-s + (0.867 + 0.497i)6-s + (−0.120 − 0.992i)7-s + (0.592 − 0.805i)8-s + (0.909 − 0.416i)9-s + (0.425 + 0.905i)10-s + (0.892 + 0.451i)11-s + (−0.310 − 0.950i)12-s + (−0.0552 + 0.998i)13-s + (−0.576 + 0.816i)14-s + (0.983 + 0.181i)15-s + (−0.979 + 0.200i)16-s + (−0.945 + 0.325i)17-s + ⋯
L(s)  = 1  + (−0.741 − 0.670i)2-s + (−0.977 + 0.212i)3-s + (0.100 + 0.994i)4-s + (−0.922 − 0.386i)5-s + (0.867 + 0.497i)6-s + (−0.120 − 0.992i)7-s + (0.592 − 0.805i)8-s + (0.909 − 0.416i)9-s + (0.425 + 0.905i)10-s + (0.892 + 0.451i)11-s + (−0.310 − 0.950i)12-s + (−0.0552 + 0.998i)13-s + (−0.576 + 0.816i)14-s + (0.983 + 0.181i)15-s + (−0.979 + 0.200i)16-s + (−0.945 + 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.703 + 0.710i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.703 + 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2938684153 + 0.1226431891i\)
\(L(\frac12)\) \(\approx\) \(0.2938684153 + 0.1226431891i\)
\(L(1)\) \(\approx\) \(0.4257946783 - 0.09927378057i\)
\(L(1)\) \(\approx\) \(0.4257946783 - 0.09927378057i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.741 - 0.670i)T \)
3 \( 1 + (-0.977 + 0.212i)T \)
5 \( 1 + (-0.922 - 0.386i)T \)
7 \( 1 + (-0.120 - 0.992i)T \)
11 \( 1 + (0.892 + 0.451i)T \)
13 \( 1 + (-0.0552 + 0.998i)T \)
17 \( 1 + (-0.945 + 0.325i)T \)
19 \( 1 + (-0.285 - 0.958i)T \)
23 \( 1 + (0.389 + 0.921i)T \)
29 \( 1 + (-0.638 - 0.769i)T \)
31 \( 1 + (0.966 + 0.257i)T \)
37 \( 1 + (-0.383 - 0.923i)T \)
41 \( 1 + (0.682 + 0.730i)T \)
43 \( 1 + (-0.999 - 0.0325i)T \)
47 \( 1 + (-0.807 - 0.589i)T \)
53 \( 1 + (-0.247 - 0.968i)T \)
59 \( 1 + (-0.197 - 0.980i)T \)
61 \( 1 + (-0.974 + 0.225i)T \)
67 \( 1 + (-0.260 + 0.965i)T \)
71 \( 1 + (0.951 + 0.307i)T \)
73 \( 1 + (-0.247 + 0.968i)T \)
79 \( 1 + (0.999 + 0.0260i)T \)
83 \( 1 + (0.929 - 0.368i)T \)
89 \( 1 + (-0.705 - 0.708i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−22.15038472951954513925776456346, −20.73020341975810156651541831285, −19.67796719888091619564467625261, −19.03387734227420216025580868452, −18.4066570113400003719304794204, −17.83845444675193741335936689163, −16.79243445425913527536982562203, −16.30950102045544668110985621422, −15.28714659123467387074203568870, −15.085108272896480375372705422245, −13.84378531331502630897964366555, −12.55695722652412115775023031513, −11.9145356914428582108810076183, −11.04868824761283762344583335875, −10.514997993031152099837503772927, −9.3695620103830083271366416013, −8.44857739041446593499745659089, −7.759606534916004091051884806306, −6.64414699129286209338341019877, −6.27267588492832204628231660713, −5.271106642931374670221742481610, −4.36857490053955882283843198094, −2.92805906676940384772588990581, −1.56301812806214519767418254049, −0.2835492158514582397482946633, 0.87770028400019186136426546952, 1.83396915334800877203807719080, 3.574479142690291403104062693291, 4.213069531878100303751227230574, 4.76905504438443823202123287066, 6.70764749243165178903758822766, 6.91602571527809784938396303959, 7.99434518731056340898519373729, 9.1464492744108532966560616865, 9.678531762386464965580516389852, 10.80381386714492567686248711077, 11.39359294608603554774461634469, 11.82424529794820798966667400366, 12.835924822454960571173887092545, 13.48551435274813213902585562468, 14.99242249042530386286773724514, 15.89981837273328619115310471276, 16.53047011872898407621647807870, 17.32668440875764948887361436095, 17.5716058427148803729595758079, 18.88871797357932019914462630634, 19.57874032747064764635479522675, 20.02040353336134334805881838309, 21.066964918791342593888858169972, 21.69894282125030099284435569168

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