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Properties

Label 1-407-407.377-r0-0-0
Degree $1$
Conductor $407$
Sign $0.815 + 0.578i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (−0.374 − 0.927i)2-s + (−0.997 − 0.0697i)3-s + (−0.719 + 0.694i)4-s + (−0.615 − 0.788i)5-s + (0.309 + 0.951i)6-s + (0.961 + 0.275i)7-s + (0.913 + 0.406i)8-s + (0.990 + 0.139i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (−0.882 − 0.469i)13-s + (−0.104 − 0.994i)14-s + (0.559 + 0.829i)15-s + (0.0348 − 0.999i)16-s + (−0.882 + 0.469i)17-s + (−0.241 − 0.970i)18-s + ⋯
L(s)  = 1  + (−0.374 − 0.927i)2-s + (−0.997 − 0.0697i)3-s + (−0.719 + 0.694i)4-s + (−0.615 − 0.788i)5-s + (0.309 + 0.951i)6-s + (0.961 + 0.275i)7-s + (0.913 + 0.406i)8-s + (0.990 + 0.139i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (−0.882 − 0.469i)13-s + (−0.104 − 0.994i)14-s + (0.559 + 0.829i)15-s + (0.0348 − 0.999i)16-s + (−0.882 + 0.469i)17-s + (−0.241 − 0.970i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.815 + 0.578i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.815 + 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3175880542 + 0.1011391980i\)
\(L(\frac12)\) \(\approx\) \(0.3175880542 + 0.1011391980i\)
\(L(1)\) \(\approx\) \(0.4673624881 - 0.1781968783i\)
\(L(1)\) \(\approx\) \(0.4673624881 - 0.1781968783i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.374 - 0.927i)T \)
3 \( 1 + (-0.997 - 0.0697i)T \)
5 \( 1 + (-0.615 - 0.788i)T \)
7 \( 1 + (0.961 + 0.275i)T \)
13 \( 1 + (-0.882 - 0.469i)T \)
17 \( 1 + (-0.882 + 0.469i)T \)
19 \( 1 + (-0.997 - 0.0697i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.438 + 0.898i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (-0.615 + 0.788i)T \)
59 \( 1 + (0.559 + 0.829i)T \)
61 \( 1 + (0.990 - 0.139i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.374 + 0.927i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.0348 + 0.999i)T \)
83 \( 1 + (-0.882 + 0.469i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (0.669 + 0.743i)T \)
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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

−24.068467041788847964898649677030, −23.596954781257752655912385098989, −22.60323726234600251541963490821, −22.12462512531093392387363010865, −20.92010609767278233773834908293, −19.47074926596961330739252664444, −18.812308911835928698333229460779, −17.754962538506925702519979764204, −17.44573137872646979406190563868, −16.3497773772780424735314401942, −15.57247495411402261639522760061, −14.70659627880969195149611632127, −14.0037893718278221517739003345, −12.58466682144797459300253424382, −11.476831092071105783161554918233, −10.75787292697028708783741303958, −9.95908085429508209578684877623, −8.60045614858117721038454905734, −7.53588022014528130869463732364, −6.886804572470362007950108084460, −6.01507051869195097460430854526, −4.6467807861878578976767468500, −4.28428410483638441525669570122, −2.04878039437810700285580763815, −0.28906978020757794894371236523, 1.16628916289618202070755608776, 2.23776007086889480503448696456, 4.0696010563124699229201028028, 4.680673708826546019768582164343, 5.62097658870658687164958955823, 7.32800566598940470918622182197, 8.14494944296992451007074097509, 9.096296214603339727858151613767, 10.26415095008456488333782552576, 11.1795622473836567375861662647, 11.7616447500156709170200545364, 12.59057252938526047357009169561, 13.22484922743391150629211931943, 14.79364127950893110278266714350, 15.81655014971213325263295187933, 16.97875761116841965145666916985, 17.415223007151739455236462120363, 18.22746822146663872026153733352, 19.29049049548279756200370577988, 20.00912987967658064502997003833, 21.01381906080151562024166740905, 21.72324331270152303019297404911, 22.47225054752502443059280332971, 23.599851922149248257885053945629, 24.10867394995654347353027100589

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