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Properties

Label 1-1312-1312.1197-r0-0-0
Degree $1$
Conductor $1312$
Sign $-0.973 - 0.228i$
Analytic cond. $6.09290$
Root an. cond. $6.09290$
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  + (0.707 − 0.707i)3-s + (0.156 − 0.987i)5-s + (0.309 − 0.951i)7-s − i·9-s + (0.156 + 0.987i)11-s + (−0.891 − 0.453i)13-s + (−0.587 − 0.809i)15-s + (0.587 − 0.809i)17-s + (−0.453 − 0.891i)19-s + (−0.453 − 0.891i)21-s + (−0.951 + 0.309i)23-s + (−0.951 − 0.309i)25-s + (−0.707 − 0.707i)27-s + (−0.156 + 0.987i)29-s + (−0.809 − 0.587i)31-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (0.156 − 0.987i)5-s + (0.309 − 0.951i)7-s − i·9-s + (0.156 + 0.987i)11-s + (−0.891 − 0.453i)13-s + (−0.587 − 0.809i)15-s + (0.587 − 0.809i)17-s + (−0.453 − 0.891i)19-s + (−0.453 − 0.891i)21-s + (−0.951 + 0.309i)23-s + (−0.951 − 0.309i)25-s + (−0.707 − 0.707i)27-s + (−0.156 + 0.987i)29-s + (−0.809 − 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1312\)    =    \(2^{5} \cdot 41\)
Sign: $-0.973 - 0.228i$
Analytic conductor: \(6.09290\)
Root analytic conductor: \(6.09290\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1312} (1197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1312,\ (0:\ ),\ -0.973 - 0.228i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1876011438 - 1.616987834i\)
\(L(\frac12)\) \(\approx\) \(0.1876011438 - 1.616987834i\)
\(L(1)\) \(\approx\) \(0.9986319901 - 0.7626723142i\)
\(L(1)\) \(\approx\) \(0.9986319901 - 0.7626723142i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.156 - 0.987i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.156 + 0.987i)T \)
13 \( 1 + (-0.891 - 0.453i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.453 - 0.891i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (-0.156 + 0.987i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.156 + 0.987i)T \)
43 \( 1 + (0.891 + 0.453i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.987 + 0.156i)T \)
59 \( 1 + (0.891 + 0.453i)T \)
61 \( 1 + (0.891 - 0.453i)T \)
67 \( 1 + (0.156 - 0.987i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 - iT \)
79 \( 1 - iT \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
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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

−21.41279186911889333380312135940, −20.81985527412874627823476022630, −19.47560963219756769882682183957, −19.19532417096058904725971465769, −18.50661978609912144867919094678, −17.534771517574921582484129269433, −16.56877500869210545524341635579, −15.902267451480084624777882899136, −14.94298548481530490473653465446, −14.47612022060576959845033913861, −14.083157559006760264481152868889, −12.88394481601466402321605739600, −11.89354624032619235787679930474, −11.13688671011845693131375517758, −10.262737326366832878030615129906, −9.72189789660427308869511415709, −8.67955901094516053901114664000, −8.13990298434269387128871304651, −7.18543002109231799780638471177, −5.95516080515292203005896649743, −5.495450772663876883289244348465, −4.109425217189130360266627133660, −3.50984228057541025038910704674, −2.43480644996171666432970973767, −1.95463928035075890483976468392, 0.53703235134791458287051231186, 1.5433729488156055191675323683, 2.34176453719970682040450500088, 3.54080091683399611060232007616, 4.51741847946695456965304154947, 5.185645526441725753911085201351, 6.47369932905346981517971946716, 7.48865870559270007379829777127, 7.69615136753734115848997273380, 8.828350803688884937703932664381, 9.56916825465659031055732468, 10.19873572097147256677564517328, 11.5587600170517838171685619642, 12.28506566656872762063751390709, 12.968755128344396320443845232149, 13.56640913010285254276145154048, 14.43426131156836205859940971259, 15.01149213564945032447591369022, 16.10509544231987345978165707484, 16.958654685801809032318829429382, 17.65048294477612463036172574769, 18.12322669421100154366803330234, 19.42741418935268465989068917975, 19.9240253652652912431135004104, 20.43278751731664150552135986534

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