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Properties

Label 1-457-457.343-r0-0-0
Degree $1$
Conductor $457$
Sign $0.237 - 0.971i$
Analytic cond. $2.12229$
Root an. cond. $2.12229$
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  + (0.789 − 0.614i)2-s + (0.945 + 0.324i)3-s + (0.245 − 0.969i)4-s + (0.677 − 0.735i)5-s + (0.945 − 0.324i)6-s + (−0.0825 − 0.996i)7-s + (−0.401 − 0.915i)8-s + (0.789 + 0.614i)9-s + (0.0825 − 0.996i)10-s + (0.401 + 0.915i)11-s + (0.546 − 0.837i)12-s + (−0.245 + 0.969i)13-s + (−0.677 − 0.735i)14-s + (0.879 − 0.475i)15-s + (−0.879 − 0.475i)16-s + (−0.0825 − 0.996i)17-s + ⋯
L(s)  = 1  + (0.789 − 0.614i)2-s + (0.945 + 0.324i)3-s + (0.245 − 0.969i)4-s + (0.677 − 0.735i)5-s + (0.945 − 0.324i)6-s + (−0.0825 − 0.996i)7-s + (−0.401 − 0.915i)8-s + (0.789 + 0.614i)9-s + (0.0825 − 0.996i)10-s + (0.401 + 0.915i)11-s + (0.546 − 0.837i)12-s + (−0.245 + 0.969i)13-s + (−0.677 − 0.735i)14-s + (0.879 − 0.475i)15-s + (−0.879 − 0.475i)16-s + (−0.0825 − 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 457 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 457 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(457\)
Sign: $0.237 - 0.971i$
Analytic conductor: \(2.12229\)
Root analytic conductor: \(2.12229\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{457} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 457,\ (0:\ ),\ 0.237 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.457544519 - 1.929589720i\)
\(L(\frac12)\) \(\approx\) \(2.457544519 - 1.929589720i\)
\(L(1)\) \(\approx\) \(2.045297549 - 0.9973646980i\)
\(L(1)\) \(\approx\) \(2.045297549 - 0.9973646980i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad457 \( 1 \)
good2 \( 1 + (0.789 - 0.614i)T \)
3 \( 1 + (0.945 + 0.324i)T \)
5 \( 1 + (0.677 - 0.735i)T \)
7 \( 1 + (-0.0825 - 0.996i)T \)
11 \( 1 + (0.401 + 0.915i)T \)
13 \( 1 + (-0.245 + 0.969i)T \)
17 \( 1 + (-0.0825 - 0.996i)T \)
19 \( 1 + (-0.0825 + 0.996i)T \)
23 \( 1 + (-0.789 + 0.614i)T \)
29 \( 1 + (0.546 + 0.837i)T \)
31 \( 1 + (0.0825 - 0.996i)T \)
37 \( 1 + (-0.945 + 0.324i)T \)
41 \( 1 + (0.0825 - 0.996i)T \)
43 \( 1 + (-0.789 + 0.614i)T \)
47 \( 1 + (-0.986 - 0.164i)T \)
53 \( 1 + (-0.945 - 0.324i)T \)
59 \( 1 + (0.0825 + 0.996i)T \)
61 \( 1 + (0.879 + 0.475i)T \)
67 \( 1 + (0.789 + 0.614i)T \)
71 \( 1 + (0.401 - 0.915i)T \)
73 \( 1 + (-0.401 + 0.915i)T \)
79 \( 1 + (0.245 - 0.969i)T \)
83 \( 1 + (0.401 - 0.915i)T \)
89 \( 1 + (0.986 - 0.164i)T \)
97 \( 1 + (-0.546 - 0.837i)T \)
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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

−24.489922154706960358326249903, −23.36759387038379795823244987204, −22.13756909160429544015160206388, −21.78932131325990861725288213218, −21.05002008338694041518067729575, −19.850808700698831252227712466893, −19.01271923506136564234954805768, −17.9954558550649452793804871720, −17.37926120685338702719350278873, −15.92387712361814274436128709988, −15.22641133590488603568571182872, −14.53766789350374625095089572900, −13.82112269669625361302957705245, −12.99327551727426234205527319097, −12.231609488869485828523374479652, −11.00837975364736641386703622483, −9.7442480635618540937931252560, −8.59110821397153651029489420087, −8.08245613519988627793707684991, −6.669493613279367290867660167476, −6.21889230764762889421520331030, −5.09501299444617310980558860897, −3.563518592463094658317995549484, −2.87373882413409533822828400765, −2.01565748522806906747991090527, 1.45404940159668745742775060658, 2.11572815850735028150458478446, 3.532799422424861862028496879480, 4.367378826106932313962745606950, 5.034270719493180099761693149333, 6.52263018544086432622223968457, 7.47155115495456893373742620122, 8.914522679491027004491051826952, 9.870591719319420010231552118559, 10.08890076182710945417952414587, 11.60599240688958629256096111999, 12.55243731115330568004250007731, 13.460332266927865628967652764188, 14.061282177353797320534765570482, 14.62094525158752238311359719283, 15.92025568565171136581518380370, 16.59547740634726040159368467929, 17.83281298239095428887955225417, 19.02891709673240066168076543843, 19.89814613378350747329073835603, 20.48854253733179259869618107665, 20.965263290867976420278025828333, 21.891317261358672816526236896216, 22.75662184170680274733545153885, 23.813064935385983851265973841961

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