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Properties

Label 1-403-403.383-r0-0-0
Degree $1$
Conductor $403$
Sign $0.931 - 0.364i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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  + (0.406 − 0.913i)2-s + (−0.913 + 0.406i)3-s + (−0.669 − 0.743i)4-s + (0.866 + 0.5i)5-s + i·6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (0.669 − 0.743i)9-s + (0.809 − 0.587i)10-s + (−0.951 − 0.309i)11-s + (0.913 + 0.406i)12-s + (0.669 − 0.743i)14-s + (−0.994 − 0.104i)15-s + (−0.104 + 0.994i)16-s + (0.309 + 0.951i)17-s + (−0.406 − 0.913i)18-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (−0.913 + 0.406i)3-s + (−0.669 − 0.743i)4-s + (0.866 + 0.5i)5-s + i·6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (0.669 − 0.743i)9-s + (0.809 − 0.587i)10-s + (−0.951 − 0.309i)11-s + (0.913 + 0.406i)12-s + (0.669 − 0.743i)14-s + (−0.994 − 0.104i)15-s + (−0.104 + 0.994i)16-s + (0.309 + 0.951i)17-s + (−0.406 − 0.913i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.931 - 0.364i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ 0.931 - 0.364i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.339910310 - 0.2526608848i\)
\(L(\frac12)\) \(\approx\) \(1.339910310 - 0.2526608848i\)
\(L(1)\) \(\approx\) \(1.103010219 - 0.2658271224i\)
\(L(1)\) \(\approx\) \(1.103010219 - 0.2658271224i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.406 - 0.913i)T \)
3 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.587 + 0.809i)T \)
23 \( 1 + (0.669 - 0.743i)T \)
29 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (0.587 - 0.809i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.743 - 0.669i)T \)
73 \( 1 + (0.743 - 0.669i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (0.994 - 0.104i)T \)
89 \( 1 + (0.207 + 0.978i)T \)
97 \( 1 + (0.743 - 0.669i)T \)
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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

−24.26351515772182840785073176201, −23.722290171478277639651707707047, −22.902393292749646443140210553399, −21.940314398137556077513134470216, −21.19978510193286260087561084516, −20.43731745419207001264546941409, −18.62896790090387556580716620273, −17.87406728693477949334394129239, −17.480767169199018393871888862559, −16.56246523011158289527061423511, −15.82098613559760972066097358630, −14.65347397319469495789881355164, −13.53592197697811584871520155116, −13.18628171326112432187643871188, −12.08800341569481220576212014587, −11.12517732949644811354107156233, −9.91193030842852275461909501542, −8.82690529357086045793073825465, −7.53089096169126827157476724860, −7.10238350612462463310410245920, −5.5112845980149559322505192056, −5.35461350203824026699719341532, −4.38365829360658329319934437145, −2.47646239494388218680033108691, −0.95823890758695230061606527233, 1.24966126269426858726965417751, 2.35541302534561704343657273545, 3.62017940260752069103865404338, 4.90335552519675430505576317196, 5.54127456765346483706142096191, 6.32222771607887907745805203291, 8.01254333327104701198339063078, 9.34888524311789793797052479981, 10.262938117714538053918248009009, 10.833703145473795137283096186006, 11.59756271430191227703371045599, 12.639635925093622485690782551, 13.45659224328985088693485091644, 14.65766602390149854836026977081, 15.11244091697333675115402781339, 16.60529042521665054819442081506, 17.5271680308193887794170878263, 18.428063200616631480102947672288, 18.66393148607754815556710125773, 20.39717929990740312627596447568, 21.19533630420087661978856057696, 21.5107071692566101266196615958, 22.40068998130319745686459792579, 23.21127329573144442602647387895, 24.00983746178726599544605471208

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