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Properties

Label 1-449-449.194-r0-0-0
Degree $1$
Conductor $449$
Sign $0.956 - 0.291i$
Analytic cond. $2.08514$
Root an. cond. $2.08514$
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  + (−0.111 − 0.993i)2-s + (−0.998 + 0.0560i)3-s + (−0.974 + 0.222i)4-s + (0.623 + 0.781i)5-s + (0.167 + 0.985i)6-s + (−0.781 − 0.623i)7-s + (0.330 + 0.943i)8-s + (0.993 − 0.111i)9-s + (0.707 − 0.707i)10-s + (−0.623 − 0.781i)11-s + (0.960 − 0.276i)12-s + (−0.985 + 0.167i)13-s + (−0.532 + 0.846i)14-s + (−0.666 − 0.745i)15-s + (0.900 − 0.433i)16-s + (−0.815 + 0.578i)17-s + ⋯
L(s)  = 1  + (−0.111 − 0.993i)2-s + (−0.998 + 0.0560i)3-s + (−0.974 + 0.222i)4-s + (0.623 + 0.781i)5-s + (0.167 + 0.985i)6-s + (−0.781 − 0.623i)7-s + (0.330 + 0.943i)8-s + (0.993 − 0.111i)9-s + (0.707 − 0.707i)10-s + (−0.623 − 0.781i)11-s + (0.960 − 0.276i)12-s + (−0.985 + 0.167i)13-s + (−0.532 + 0.846i)14-s + (−0.666 − 0.745i)15-s + (0.900 − 0.433i)16-s + (−0.815 + 0.578i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $0.956 - 0.291i$
Analytic conductor: \(2.08514\)
Root analytic conductor: \(2.08514\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (0:\ ),\ 0.956 - 0.291i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6431290223 - 0.09583249641i\)
\(L(\frac12)\) \(\approx\) \(0.6431290223 - 0.09583249641i\)
\(L(1)\) \(\approx\) \(0.6191273179 - 0.1970225344i\)
\(L(1)\) \(\approx\) \(0.6191273179 - 0.1970225344i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (-0.111 - 0.993i)T \)
3 \( 1 + (-0.998 + 0.0560i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (-0.781 - 0.623i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (-0.985 + 0.167i)T \)
17 \( 1 + (-0.815 + 0.578i)T \)
19 \( 1 + (0.875 - 0.483i)T \)
23 \( 1 + (0.974 + 0.222i)T \)
29 \( 1 + (-0.276 - 0.960i)T \)
31 \( 1 + (-0.276 + 0.960i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (0.846 + 0.532i)T \)
43 \( 1 + (0.960 + 0.276i)T \)
47 \( 1 + (0.167 + 0.985i)T \)
53 \( 1 + (0.846 + 0.532i)T \)
59 \( 1 + (0.532 - 0.846i)T \)
61 \( 1 + (0.111 - 0.993i)T \)
67 \( 1 + T \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (-0.875 + 0.483i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (0.998 - 0.0560i)T \)
89 \( 1 + (-0.433 + 0.900i)T \)
97 \( 1 + (-0.330 + 0.943i)T \)
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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

−24.32316053278043527195732389134, −23.1623727795795304571515233589, −22.46618690467315615725144579063, −21.91874942628517966374088208791, −20.79287232983157386999374745043, −19.55722581370868466179035087756, −18.37499358012309196209776280244, −17.90299341593427196615611800922, −17.02838627284942633609026653632, −16.30055327259325276404687358040, −15.68425322578192639502131793903, −14.6997977381668394674586435351, −13.28012800954167005361538692961, −12.81151589739540715279453015711, −12.03925710886663196194308036761, −10.435412379082890634699678790817, −9.59775504284893712946122063785, −9.032860265525312701530256883637, −7.49928617745247065750042116425, −6.8380451978332084587830843002, −5.55638320911084162155690686940, −5.32535549120348283196969406042, −4.25586590314982636746469726814, −2.2860985301627268429471923521, −0.59573481559068490220409909079, 0.90749632206050417292244819554, 2.43071916455046386611621980017, 3.373144017695374616713257315579, 4.61074075045477469937272915994, 5.62072263561081811305573455991, 6.65789009693083047142285465227, 7.63056244138781253697772384330, 9.35112051138081125801341737475, 9.91448672987136570125612880928, 10.874875844948508107407785468155, 11.2162655503078281235370845826, 12.53979974552771695931736651262, 13.248591216223104339444685169197, 13.96970012354869375744973752944, 15.30379864340203641184332957616, 16.49049925341707517124625649498, 17.341999774923271812320335003438, 17.88650704238482004638431443609, 18.93963869833903921407339170290, 19.426246733159778139363731156674, 20.69168687851989536148866066089, 21.70588444024079067046314954325, 22.05394726607195080892346524452, 22.84211498374283125288148510886, 23.59188295853804796315842945324

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