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Properties

Label 1-896-896.867-r0-0-0
Degree $1$
Conductor $896$
Sign $-0.671 + 0.740i$
Analytic cond. $4.16100$
Root an. cond. $4.16100$
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  + (−0.980 − 0.195i)3-s + (−0.555 + 0.831i)5-s + (0.923 + 0.382i)9-s + (0.195 + 0.980i)11-s + (0.555 + 0.831i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.831 + 0.555i)19-s + (−0.382 + 0.923i)23-s + (−0.382 − 0.923i)25-s + (−0.831 − 0.555i)27-s + (0.195 − 0.980i)29-s − i·31-s − i·33-s + (0.831 + 0.555i)37-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)3-s + (−0.555 + 0.831i)5-s + (0.923 + 0.382i)9-s + (0.195 + 0.980i)11-s + (0.555 + 0.831i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.831 + 0.555i)19-s + (−0.382 + 0.923i)23-s + (−0.382 − 0.923i)25-s + (−0.831 − 0.555i)27-s + (0.195 − 0.980i)29-s − i·31-s − i·33-s + (0.831 + 0.555i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.671 + 0.740i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ -0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2966904685 + 0.6693229723i\)
\(L(\frac12)\) \(\approx\) \(0.2966904685 + 0.6693229723i\)
\(L(1)\) \(\approx\) \(0.6715673243 + 0.2472327047i\)
\(L(1)\) \(\approx\) \(0.6715673243 + 0.2472327047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.980 - 0.195i)T \)
5 \( 1 + (-0.555 + 0.831i)T \)
11 \( 1 + (0.195 + 0.980i)T \)
13 \( 1 + (0.555 + 0.831i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (-0.831 + 0.555i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (0.195 - 0.980i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.831 + 0.555i)T \)
41 \( 1 + (0.382 - 0.923i)T \)
43 \( 1 + (0.980 - 0.195i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.195 + 0.980i)T \)
59 \( 1 + (-0.555 + 0.831i)T \)
61 \( 1 + (-0.980 - 0.195i)T \)
67 \( 1 + (-0.980 - 0.195i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (0.831 - 0.555i)T \)
89 \( 1 + (-0.382 - 0.923i)T \)
97 \( 1 - iT \)
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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

−21.62576576292608941834584396514, −21.00666527998996751182213841035, −20.1340790447037787877216254219, −19.29270142816045384500046324581, −18.355901655966127371921551942129, −17.68844106413351765714786520010, −16.56551313146703925340703878021, −16.35039351832326951103780087791, −15.57176505928702496158249564278, −14.55699812769350766215164112337, −13.37852288177987032028748354801, −12.61397475612236348210514506509, −12.00105067824307132999126120225, −11.04725208915142336562205685045, −10.54950937340418950987638630620, −9.31542439297430103046284072879, −8.535770994523907982275273173384, −7.63869533635001276162702962441, −6.512002007291310470170429181253, −5.67876387309237578706940006202, −4.90767673874091664833822496384, −4.04383053335849467556927069951, −3.035297542007409471110064547033, −1.246684761958857448632943030504, −0.43830365295492802205698099378, 1.36415841740343494259727773596, 2.37901231895029041828920519292, 4.00154489500898366813622949007, 4.28280022924751240903453869546, 5.87177505507086018251155645765, 6.29729953574686437031286183169, 7.37754941861496277809192151217, 7.86827088426668482412487463727, 9.3465209060616553917331742952, 10.26676811955441936520354189871, 10.897162692793643509615143992373, 11.82582752674315498370711343129, 12.243702548302152696806468278921, 13.31550011987740834228351286931, 14.30620817789313113038485541955, 15.206152809593998952797063452416, 15.801729585654728812236326964714, 16.881167611357043917359288857627, 17.38383074546460944153953564369, 18.40409179255965287321606881268, 18.91056818893781615038154610918, 19.61163706953782546714949610257, 20.86182353544190779985670497302, 21.57630432510717446829622649898, 22.4160935437094681286251223103

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