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Properties

Label 1-967-967.36-r0-0-0
Degree $1$
Conductor $967$
Sign $0.0853 + 0.996i$
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  + (0.754 + 0.655i)2-s + (0.353 − 0.935i)3-s + (0.139 + 0.990i)4-s + (−0.0941 + 0.995i)5-s + (0.880 − 0.474i)6-s + (0.0357 − 0.999i)7-s + (−0.544 + 0.838i)8-s + (−0.750 − 0.660i)9-s + (−0.724 + 0.689i)10-s + (−0.477 + 0.878i)11-s + (0.975 + 0.219i)12-s + (0.999 + 0.0260i)13-s + (0.682 − 0.730i)14-s + (0.898 + 0.439i)15-s + (−0.961 + 0.276i)16-s + (0.987 − 0.155i)17-s + ⋯
L(s)  = 1  + (0.754 + 0.655i)2-s + (0.353 − 0.935i)3-s + (0.139 + 0.990i)4-s + (−0.0941 + 0.995i)5-s + (0.880 − 0.474i)6-s + (0.0357 − 0.999i)7-s + (−0.544 + 0.838i)8-s + (−0.750 − 0.660i)9-s + (−0.724 + 0.689i)10-s + (−0.477 + 0.878i)11-s + (0.975 + 0.219i)12-s + (0.999 + 0.0260i)13-s + (0.682 − 0.730i)14-s + (0.898 + 0.439i)15-s + (−0.961 + 0.276i)16-s + (0.987 − 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.0853 + 0.996i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.0853 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.707380520 + 1.567393610i\)
\(L(\frac12)\) \(\approx\) \(1.707380520 + 1.567393610i\)
\(L(1)\) \(\approx\) \(1.545587732 + 0.6067622379i\)
\(L(1)\) \(\approx\) \(1.545587732 + 0.6067622379i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.754 + 0.655i)T \)
3 \( 1 + (0.353 - 0.935i)T \)
5 \( 1 + (-0.0941 + 0.995i)T \)
7 \( 1 + (0.0357 - 0.999i)T \)
11 \( 1 + (-0.477 + 0.878i)T \)
13 \( 1 + (0.999 + 0.0260i)T \)
17 \( 1 + (0.987 - 0.155i)T \)
19 \( 1 + (-0.488 + 0.872i)T \)
23 \( 1 + (-0.442 + 0.896i)T \)
29 \( 1 + (0.0487 + 0.998i)T \)
31 \( 1 + (0.969 - 0.244i)T \)
37 \( 1 + (0.602 + 0.797i)T \)
41 \( 1 + (0.460 - 0.887i)T \)
43 \( 1 + (0.728 + 0.684i)T \)
47 \( 1 + (0.903 - 0.428i)T \)
53 \( 1 + (-0.158 + 0.987i)T \)
59 \( 1 + (0.672 - 0.739i)T \)
61 \( 1 + (0.538 + 0.842i)T \)
67 \( 1 + (-0.696 + 0.717i)T \)
71 \( 1 + (-0.945 - 0.325i)T \)
73 \( 1 + (-0.158 - 0.987i)T \)
79 \( 1 + (-0.285 + 0.958i)T \)
83 \( 1 + (0.266 + 0.963i)T \)
89 \( 1 + (-0.272 - 0.962i)T \)
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

−21.27609589426243234921137560942, −21.063582499369436048527368685157, −20.32903931385966017097988786856, −19.30441948309241054920034902311, −18.88390566599023724105009087905, −17.66000787689238379597570865022, −16.339549560520474334665187377876, −15.94270319058031604253907306630, −15.2564452790612569859654672605, −14.35756437185942233356547764081, −13.52508829236738215101429037773, −12.83155051579168427988872096052, −11.8915568928592223694296826843, −11.20096868718066384774900291857, −10.352335395152821359667051781828, −9.42525773225773666951255420232, −8.696493742492261977049241724891, −8.09934124170058264541791416725, −6.042963454350822814149854002973, −5.67697999133729999994295929123, −4.71723944708335683407478261838, −4.00895338233835105859279011823, −2.999089191897167721221235310743, −2.22560379209809587389176244762, −0.76156280560106938484022772636, 1.4310202629733105806969144333, 2.62434183962072883247936260022, 3.48272326476242123639797724091, 4.19887952178278589116160691513, 5.67772250809759750632406350204, 6.34944668200922373904553732367, 7.290831459240765363110280306912, 7.598456894907980257310114081911, 8.43214556211398915839084132132, 9.85726275898066968043033489971, 10.81916641600360484604265228610, 11.76008088693356953492351908572, 12.544095423411422643700796013259, 13.41813685586618790715371110631, 13.99297655385771326489777967018, 14.58727740606205834006878000748, 15.36864812014161165125362135191, 16.32429388339238066050771646138, 17.31743085874370813319902894855, 17.93581647103476942929308191250, 18.64944608885292969019551673238, 19.56809545092620150200943830993, 20.62836221571642883214574646582, 20.95567551761406476085780853635, 22.28437116397627403205572959884

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