Properties

Label 1-847-847.96-r0-0-0
Degree $1$
Conductor $847$
Sign $0.638 + 0.769i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (−0.997 − 0.0760i)2-s + (0.104 + 0.994i)3-s + (0.988 + 0.151i)4-s + (−0.123 − 0.992i)5-s + (−0.0285 − 0.999i)6-s + (−0.974 − 0.226i)8-s + (−0.978 + 0.207i)9-s + (0.0475 + 0.998i)10-s + (−0.0475 + 0.998i)12-s + (0.774 − 0.633i)13-s + (0.974 − 0.226i)15-s + (0.953 + 0.299i)16-s + (−0.0665 + 0.997i)17-s + (0.991 − 0.132i)18-s + (−0.532 + 0.846i)19-s + (0.0285 − 0.999i)20-s + ⋯
L(s)  = 1  + (−0.997 − 0.0760i)2-s + (0.104 + 0.994i)3-s + (0.988 + 0.151i)4-s + (−0.123 − 0.992i)5-s + (−0.0285 − 0.999i)6-s + (−0.974 − 0.226i)8-s + (−0.978 + 0.207i)9-s + (0.0475 + 0.998i)10-s + (−0.0475 + 0.998i)12-s + (0.774 − 0.633i)13-s + (0.974 − 0.226i)15-s + (0.953 + 0.299i)16-s + (−0.0665 + 0.997i)17-s + (0.991 − 0.132i)18-s + (−0.532 + 0.846i)19-s + (0.0285 − 0.999i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (96, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.638 + 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7617297351 + 0.3576959910i\)
\(L(\frac12)\) \(\approx\) \(0.7617297351 + 0.3576959910i\)
\(L(1)\) \(\approx\) \(0.6946906007 + 0.1360832935i\)
\(L(1)\) \(\approx\) \(0.6946906007 + 0.1360832935i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.997 - 0.0760i)T \)
3 \( 1 + (0.104 + 0.994i)T \)
5 \( 1 + (-0.123 - 0.992i)T \)
13 \( 1 + (0.774 - 0.633i)T \)
17 \( 1 + (-0.0665 + 0.997i)T \)
19 \( 1 + (-0.532 + 0.846i)T \)
23 \( 1 + (0.580 - 0.814i)T \)
29 \( 1 + (-0.696 + 0.717i)T \)
31 \( 1 + (-0.964 - 0.263i)T \)
37 \( 1 + (0.449 + 0.893i)T \)
41 \( 1 + (0.941 - 0.336i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.991 + 0.132i)T \)
53 \( 1 + (0.953 - 0.299i)T \)
59 \( 1 + (0.761 - 0.647i)T \)
61 \( 1 + (-0.432 + 0.901i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (-0.985 + 0.170i)T \)
73 \( 1 + (0.749 + 0.662i)T \)
79 \( 1 + (-0.483 + 0.875i)T \)
83 \( 1 + (0.993 + 0.113i)T \)
89 \( 1 + (0.786 - 0.618i)T \)
97 \( 1 + (0.921 - 0.389i)T \)
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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

−21.95250753061475515320692599608, −20.98826344901993329814995413471, −20.08783917779890451930296403860, −19.25963569395685823866283268680, −18.82981085289983891796392099084, −18.06574103370080104614500403121, −17.5690408195077360205980948099, −16.56667942349368121736172724776, −15.62112505529853835653079454431, −14.81192856893254927099469505791, −13.96719965315344741754389631913, −13.11579133340847663358090421542, −11.89967938786942876668365485771, −11.261333106878365080231600739456, −10.741684310624545920299366187921, −9.34237947232015479793728072920, −8.86120727600716598516400408739, −7.55781245874528112821467876868, −7.255294438057210821569646296674, −6.39992861303094905733203136311, −5.61487878281202597673599604083, −3.7247157004267934955276433857, −2.64561553352508346834481018236, −1.99063792511343456209231092861, −0.66835552795195481659679908488, 0.93051951771053297006634839247, 2.15152375480295519307667967301, 3.44829241923610225605898035654, 4.206081363493420925499451734154, 5.50279318386022602829870713163, 6.14489683818898350772808530062, 7.65565139950117694500301386658, 8.49580649638038109784483902435, 8.88257974105120012129070249463, 9.836866840291554566472893138880, 10.66006789606774623603541251035, 11.22496022151622284183901410902, 12.40596721083102162180984544254, 13.03826931780789310793436478112, 14.5480683405905943912756391996, 15.22087297989017446616492810415, 16.04086890788754756399513854278, 16.67593357650285820743428146838, 17.15877066852873905638408983178, 18.171130116232210195303866413566, 19.15764678206377763769610281776, 19.97198481695012498633650233860, 20.61610155301698593150039896767, 21.043418159172547793919922303388, 21.9299680533686658388210920163

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