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Properties

Label 2-56-56.53-c1-0-5
Degree $2$
Conductor $56$
Sign $-0.599 + 0.800i$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.242 − 1.39i)2-s + (−2.13 − 1.22i)3-s + (−1.88 + 0.674i)4-s + (1.28 − 0.742i)5-s + (−1.19 + 3.26i)6-s + (0.129 − 2.64i)7-s + (1.39 + 2.45i)8-s + (1.52 + 2.64i)9-s + (−1.34 − 1.61i)10-s + (4.37 + 2.52i)11-s + (4.84 + 0.878i)12-s − 2.58i·13-s + (−3.71 + 0.459i)14-s − 3.65·15-s + (3.08 − 2.54i)16-s + (−0.629 + 1.09i)17-s + ⋯
L(s)  = 1  + (−0.171 − 0.985i)2-s + (−1.22 − 0.710i)3-s + (−0.941 + 0.337i)4-s + (0.575 − 0.332i)5-s + (−0.489 + 1.33i)6-s + (0.0490 − 0.998i)7-s + (0.493 + 0.869i)8-s + (0.508 + 0.880i)9-s + (−0.425 − 0.510i)10-s + (1.31 + 0.760i)11-s + (1.39 + 0.253i)12-s − 0.717i·13-s + (−0.992 + 0.122i)14-s − 0.943·15-s + (0.772 − 0.635i)16-s + (−0.152 + 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.599 + 0.800i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ -0.599 + 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.260545 - 0.521080i\)
\(L(\frac12)\) \(\approx\) \(0.260545 - 0.521080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.242 + 1.39i)T \)
7 \( 1 + (-0.129 + 2.64i)T \)
good3 \( 1 + (2.13 + 1.22i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.28 + 0.742i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.37 - 2.52i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.58iT - 13T^{2} \)
17 \( 1 + (0.629 - 1.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.68 - 1.54i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.697 - 1.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.638iT - 29T^{2} \)
31 \( 1 + (-1.82 + 3.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.21 + 3.01i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.36T + 41T^{2} \)
43 \( 1 + 1.02iT - 43T^{2} \)
47 \( 1 + (-5.48 - 9.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.99 + 2.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.01 - 1.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.1 - 6.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.443 + 0.256i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.41T + 71T^{2} \)
73 \( 1 + (4.94 - 8.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.35 + 7.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.97iT - 83T^{2} \)
89 \( 1 + (-1.29 - 2.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.57T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.51070269117649338975347692414, −13.22917084105403462517219365502, −12.55441080659480700043459456589, −11.49068222038303226610359570007, −10.51789614269602631827576020433, −9.349200257525703165581913937698, −7.49467039967980964434217327977, −5.97110231946454570739929281401, −4.32183072717123646166876550521, −1.32682520934264787035618537739, 4.50827714234194357453067902611, 5.90971479269990985807710879023, 6.51069553913320388531566593225, 8.749629292410079874905601897490, 9.684284134449868208117465493461, 11.02475696554353353040410043198, 12.11991329293397220395978481633, 13.81147473559183785985719821243, 14.81230152560918814625734736010, 15.90683368546294186318999136781

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