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Properties

Label 1-2312-2312.261-r1-0-0
Degree $1$
Conductor $2312$
Sign $0.842 + 0.539i$
Analytic cond. $248.458$
Root an. cond. $248.458$
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  + (0.978 − 0.206i)3-s + (−0.884 + 0.466i)5-s + (−0.690 + 0.723i)7-s + (0.914 − 0.403i)9-s + (0.0692 + 0.997i)11-s + (0.995 + 0.0922i)13-s + (−0.769 + 0.638i)15-s + (0.973 + 0.228i)19-s + (−0.526 + 0.850i)21-s + (0.723 + 0.690i)23-s + (0.565 − 0.824i)25-s + (0.811 − 0.584i)27-s + (0.656 − 0.754i)29-s + (0.295 + 0.955i)31-s + (0.273 + 0.961i)33-s + ⋯
L(s)  = 1  + (0.978 − 0.206i)3-s + (−0.884 + 0.466i)5-s + (−0.690 + 0.723i)7-s + (0.914 − 0.403i)9-s + (0.0692 + 0.997i)11-s + (0.995 + 0.0922i)13-s + (−0.769 + 0.638i)15-s + (0.973 + 0.228i)19-s + (−0.526 + 0.850i)21-s + (0.723 + 0.690i)23-s + (0.565 − 0.824i)25-s + (0.811 − 0.584i)27-s + (0.656 − 0.754i)29-s + (0.295 + 0.955i)31-s + (0.273 + 0.961i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.842 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.842 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.842 + 0.539i$
Analytic conductor: \(248.458\)
Root analytic conductor: \(248.458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2312,\ (1:\ ),\ 0.842 + 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.136417000 + 0.9177796632i\)
\(L(\frac12)\) \(\approx\) \(3.136417000 + 0.9177796632i\)
\(L(1)\) \(\approx\) \(1.420913440 + 0.2048963359i\)
\(L(1)\) \(\approx\) \(1.420913440 + 0.2048963359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.978 - 0.206i)T \)
5 \( 1 + (-0.884 + 0.466i)T \)
7 \( 1 + (-0.690 + 0.723i)T \)
11 \( 1 + (0.0692 + 0.997i)T \)
13 \( 1 + (0.995 + 0.0922i)T \)
19 \( 1 + (0.973 + 0.228i)T \)
23 \( 1 + (0.723 + 0.690i)T \)
29 \( 1 + (0.656 - 0.754i)T \)
31 \( 1 + (0.295 + 0.955i)T \)
37 \( 1 + (-0.115 - 0.993i)T \)
41 \( 1 + (0.545 - 0.837i)T \)
43 \( 1 + (0.990 + 0.138i)T \)
47 \( 1 + (-0.361 - 0.932i)T \)
53 \( 1 + (-0.403 - 0.914i)T \)
59 \( 1 + (0.0461 - 0.998i)T \)
61 \( 1 + (-0.905 + 0.424i)T \)
67 \( 1 + (-0.850 + 0.526i)T \)
71 \( 1 + (0.999 - 0.0230i)T \)
73 \( 1 + (0.967 + 0.251i)T \)
79 \( 1 + (0.986 - 0.160i)T \)
83 \( 1 + (-0.824 - 0.565i)T \)
89 \( 1 + (-0.995 + 0.0922i)T \)
97 \( 1 + (0.0230 + 0.999i)T \)
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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

−19.56855697089511453742628793102, −18.83027882481330819999751006049, −18.27250504378073063321207310038, −16.87472705634440597900846946193, −16.39236820064621530297820641895, −15.7818790060465168211121581465, −15.25734709023344973395388477032, −14.1585518698001541767386910032, −13.66421454578517986344333641985, −13.01239554558251651990236804639, −12.2956202574078884727544135645, −11.13347873814028744312646554955, −10.74390796212917861233006773914, −9.63043075807336215776450496121, −9.02230869415190435848484014064, −8.29110078464203570297730283553, −7.71225866546681285192947409693, −6.88027234206834136947743042470, −5.9979351099018227510997576447, −4.72924275257940691097378880845, −4.13312993456414287233466402362, −3.19055668180224848362188104931, −2.99618086003570728323437926227, −1.21430073814539080133842702359, −0.74999934575849966942421809106, 0.73703041354555029052096071908, 1.86204558959542570750479045790, 2.77310782825018521383481714966, 3.45063327277013094118070575866, 4.05458808790485912660193253618, 5.14987302212886212507098213421, 6.31013138356318939811605481719, 7.01071519658779976081557771924, 7.634854313362600882809100482628, 8.44749671384528376395454835298, 9.16097560894805726109176271072, 9.80315174113842831249153870624, 10.68895715963275823740364217461, 11.688040671002242698577942187269, 12.32657580762416467635962570797, 12.93431669170656954828409832559, 13.85175206870517827844200843304, 14.47743408153131786455959282849, 15.40875019868842561009866422282, 15.63354589204641497607662711204, 16.26697788318651304836264507587, 17.70543213866598758269467983091, 18.21915091387969280447612399340, 18.96993188656728190284875020856, 19.483184000394855241296110678187

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