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Properties

Label 1-34e2-1156.307-r1-0-0
Degree $1$
Conductor $1156$
Sign $-0.746 - 0.665i$
Analytic cond. $124.229$
Root an. cond. $124.229$
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  + (0.273 + 0.961i)3-s + (−0.602 − 0.798i)5-s + (0.850 − 0.526i)7-s + (−0.850 + 0.526i)9-s + (−0.0922 − 0.995i)11-s + (−0.602 + 0.798i)13-s + (0.602 − 0.798i)15-s + (−0.739 + 0.673i)19-s + (0.739 + 0.673i)21-s + (0.850 − 0.526i)23-s + (−0.273 + 0.961i)25-s + (−0.739 − 0.673i)27-s + (0.0922 + 0.995i)29-s + (0.602 − 0.798i)31-s + (0.932 − 0.361i)33-s + ⋯
L(s)  = 1  + (0.273 + 0.961i)3-s + (−0.602 − 0.798i)5-s + (0.850 − 0.526i)7-s + (−0.850 + 0.526i)9-s + (−0.0922 − 0.995i)11-s + (−0.602 + 0.798i)13-s + (0.602 − 0.798i)15-s + (−0.739 + 0.673i)19-s + (0.739 + 0.673i)21-s + (0.850 − 0.526i)23-s + (−0.273 + 0.961i)25-s + (−0.739 − 0.673i)27-s + (0.0922 + 0.995i)29-s + (0.602 − 0.798i)31-s + (0.932 − 0.361i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.746 - 0.665i$
Analytic conductor: \(124.229\)
Root analytic conductor: \(124.229\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1156,\ (1:\ ),\ -0.746 - 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1736117871 - 0.4554777717i\)
\(L(\frac12)\) \(\approx\) \(0.1736117871 - 0.4554777717i\)
\(L(1)\) \(\approx\) \(0.9403776481 + 0.06389977988i\)
\(L(1)\) \(\approx\) \(0.9403776481 + 0.06389977988i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.273 + 0.961i)T \)
5 \( 1 + (-0.602 - 0.798i)T \)
7 \( 1 + (0.850 - 0.526i)T \)
11 \( 1 + (-0.0922 - 0.995i)T \)
13 \( 1 + (-0.602 + 0.798i)T \)
19 \( 1 + (-0.739 + 0.673i)T \)
23 \( 1 + (0.850 - 0.526i)T \)
29 \( 1 + (0.0922 + 0.995i)T \)
31 \( 1 + (0.602 - 0.798i)T \)
37 \( 1 + (0.932 + 0.361i)T \)
41 \( 1 + (-0.273 - 0.961i)T \)
43 \( 1 + (0.982 + 0.183i)T \)
47 \( 1 + (0.850 + 0.526i)T \)
53 \( 1 + (-0.850 + 0.526i)T \)
59 \( 1 + (-0.445 - 0.895i)T \)
61 \( 1 + (0.445 - 0.895i)T \)
67 \( 1 + (-0.739 + 0.673i)T \)
71 \( 1 + (0.850 - 0.526i)T \)
73 \( 1 + (-0.982 + 0.183i)T \)
79 \( 1 + (-0.739 + 0.673i)T \)
83 \( 1 + (0.273 - 0.961i)T \)
89 \( 1 + (-0.602 - 0.798i)T \)
97 \( 1 + (-0.850 + 0.526i)T \)
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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

−21.344137085131078297639579604793, −20.46070452919742857797927930541, −19.62287171171747417434063056084, −19.17208272838100254384336522561, −18.2222986752788379358262304417, −17.7215293292774417917249352947, −17.15957953921969583393160744540, −15.56389353247353770481314746435, −15.00828221842179381234225848145, −14.60995205460747710792622762557, −13.56343154741018698421662593832, −12.67149074099540271105683013735, −12.01136211336896720132042425078, −11.31701408447597529779307700173, −10.48230306304617617335233007299, −9.34994015149464311601117383750, −8.35323195368389101546689360654, −7.67486392438699377205447257418, −7.12671770622797780863003976341, −6.20796697188271666868865301141, −5.13840246936756974605115362189, −4.17579430373093270941856529881, −2.7559166383523298616303706814, −2.44358067518483824781061510932, −1.184852130623719384943822419362, 0.0993065666159135750311229512, 1.26445547283143038337042636077, 2.57096505198991501949029582553, 3.72921897587287748674347424392, 4.41825934066455943895568036020, 4.9769120983457551207266235898, 6.004420858055144791348884319206, 7.372441501594799149762848597332, 8.21084508592643548730425414308, 8.751509081373534489041520123240, 9.55269561692466506114283377593, 10.6929935875774965618403701322, 11.15647619733780120894658949522, 11.998131757633508108512827736199, 12.99590902711377569993575454252, 14.05877723219018247434052126870, 14.5110199154438942842387155515, 15.41629510745910086157117793218, 16.21947514911972601484482387309, 16.90847475495651965850830634774, 17.218762908582391995825322697409, 18.7977775724600746315735314392, 19.28714304917718451120593311549, 20.303516903021706299875140147032, 20.73712184958482532084774115027

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