Properties

Label 1-968-968.413-r1-0-0
Degree $1$
Conductor $968$
Sign $0.436 - 0.899i$
Analytic cond. $104.026$
Root an. cond. $104.026$
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  + (−0.309 − 0.951i)3-s + (−0.0855 − 0.996i)5-s + (−0.993 + 0.113i)7-s + (−0.809 + 0.587i)9-s + (−0.362 + 0.931i)13-s + (−0.921 + 0.389i)15-s + (−0.696 + 0.717i)17-s + (0.897 − 0.441i)19-s + (0.415 + 0.909i)21-s + (0.415 − 0.909i)23-s + (−0.985 + 0.170i)25-s + (0.809 + 0.587i)27-s + (−0.985 − 0.170i)29-s + (−0.998 + 0.0570i)31-s + (0.198 + 0.980i)35-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.0855 − 0.996i)5-s + (−0.993 + 0.113i)7-s + (−0.809 + 0.587i)9-s + (−0.362 + 0.931i)13-s + (−0.921 + 0.389i)15-s + (−0.696 + 0.717i)17-s + (0.897 − 0.441i)19-s + (0.415 + 0.909i)21-s + (0.415 − 0.909i)23-s + (−0.985 + 0.170i)25-s + (0.809 + 0.587i)27-s + (−0.985 − 0.170i)29-s + (−0.998 + 0.0570i)31-s + (0.198 + 0.980i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.436 - 0.899i$
Analytic conductor: \(104.026\)
Root analytic conductor: \(104.026\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 968,\ (1:\ ),\ 0.436 - 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7726713228 - 0.4839040994i\)
\(L(\frac12)\) \(\approx\) \(0.7726713228 - 0.4839040994i\)
\(L(1)\) \(\approx\) \(0.6750864942 - 0.2726686754i\)
\(L(1)\) \(\approx\) \(0.6750864942 - 0.2726686754i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.0855 - 0.996i)T \)
7 \( 1 + (-0.993 + 0.113i)T \)
13 \( 1 + (-0.362 + 0.931i)T \)
17 \( 1 + (-0.696 + 0.717i)T \)
19 \( 1 + (0.897 - 0.441i)T \)
23 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (-0.985 - 0.170i)T \)
31 \( 1 + (-0.998 + 0.0570i)T \)
37 \( 1 + (-0.774 + 0.633i)T \)
41 \( 1 + (0.564 + 0.825i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (-0.0285 - 0.999i)T \)
53 \( 1 + (0.870 + 0.491i)T \)
59 \( 1 + (0.564 - 0.825i)T \)
61 \( 1 + (0.610 + 0.791i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (0.736 + 0.676i)T \)
79 \( 1 + (-0.974 - 0.226i)T \)
83 \( 1 + (0.198 - 0.980i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.0855 - 0.996i)T \)
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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

−21.998249404306687400168079051595, −20.93917959767319688153130013059, −20.105906746897188471716146384386, −19.48907628849726632435178467072, −18.450613027943464252438545403915, −17.74821607419101787354304422504, −16.90916074565340519068302732421, −15.92010516876030608943141685629, −15.5454778337542933850508829882, −14.69601372902296433611235918197, −13.8755129546151653095119316164, −12.911080366548312474987675568407, −11.86683381408590248859372456103, −11.07533587274434185059209202497, −10.37562783982687546851940952726, −9.66365301756205964950120065565, −9.0171563157611903896741273827, −7.54758488519556904139568470849, −6.9258503797468935265349985209, −5.8002417917085672675195415781, −5.23829021768058561055205639242, −3.74507525426616571600593276053, −3.37895717776190774899061253114, −2.38548889908501666774722642236, −0.43730796864975002006988333766, 0.45096952318098556762904979568, 1.55923805049046515777474496394, 2.49777551019990131600197385799, 3.76665485430331593825751182005, 4.87349061636678768173129637934, 5.73088581428415704110521855052, 6.653430723770483746037086026, 7.281567052865839491867374386187, 8.45246026493339105308981469894, 9.05425432622142989935145243096, 9.95705159555225120878479285561, 11.252820276075378830760176925244, 11.87952850018096697981076325188, 12.85974639649769349410950331801, 13.05425436234563946455674447904, 14.00762881015623865398964933381, 15.10078692266276068072150648620, 16.21640208928653343028045334960, 16.6629090491550032365519792005, 17.37364098147550500650710702726, 18.3843429083329015957608095519, 19.065820691877477935271312691425, 19.84055575934904388359708032832, 20.29157397001843304043770802155, 21.56148807637026825540075272334

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