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Properties

Label 1-316-316.259-r1-0-0
Degree $1$
Conductor $316$
Sign $-0.303 - 0.952i$
Analytic cond. $33.9589$
Root an. cond. $33.9589$
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  + (−0.885 + 0.464i)3-s + (0.120 + 0.992i)5-s + (−0.885 + 0.464i)7-s + (0.568 − 0.822i)9-s + (−0.120 + 0.992i)11-s + (−0.748 + 0.663i)13-s + (−0.568 − 0.822i)15-s + (−0.748 + 0.663i)17-s + (0.970 + 0.239i)19-s + (0.568 − 0.822i)21-s − 23-s + (−0.970 + 0.239i)25-s + (−0.120 + 0.992i)27-s + (0.568 + 0.822i)29-s + (0.354 + 0.935i)31-s + ⋯
L(s)  = 1  + (−0.885 + 0.464i)3-s + (0.120 + 0.992i)5-s + (−0.885 + 0.464i)7-s + (0.568 − 0.822i)9-s + (−0.120 + 0.992i)11-s + (−0.748 + 0.663i)13-s + (−0.568 − 0.822i)15-s + (−0.748 + 0.663i)17-s + (0.970 + 0.239i)19-s + (0.568 − 0.822i)21-s − 23-s + (−0.970 + 0.239i)25-s + (−0.120 + 0.992i)27-s + (0.568 + 0.822i)29-s + (0.354 + 0.935i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 316 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 316 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(316\)    =    \(2^{2} \cdot 79\)
Sign: $-0.303 - 0.952i$
Analytic conductor: \(33.9589\)
Root analytic conductor: \(33.9589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{316} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 316,\ (1:\ ),\ -0.303 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2528806673 + 0.3458255982i\)
\(L(\frac12)\) \(\approx\) \(-0.2528806673 + 0.3458255982i\)
\(L(1)\) \(\approx\) \(0.5108846400 + 0.3631645091i\)
\(L(1)\) \(\approx\) \(0.5108846400 + 0.3631645091i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.885 + 0.464i)T \)
5 \( 1 + (0.120 + 0.992i)T \)
7 \( 1 + (-0.885 + 0.464i)T \)
11 \( 1 + (-0.120 + 0.992i)T \)
13 \( 1 + (-0.748 + 0.663i)T \)
17 \( 1 + (-0.748 + 0.663i)T \)
19 \( 1 + (0.970 + 0.239i)T \)
23 \( 1 - T \)
29 \( 1 + (0.568 + 0.822i)T \)
31 \( 1 + (0.354 + 0.935i)T \)
37 \( 1 + (-0.970 - 0.239i)T \)
41 \( 1 + (0.120 + 0.992i)T \)
43 \( 1 + (-0.120 - 0.992i)T \)
47 \( 1 + (0.970 - 0.239i)T \)
53 \( 1 + (0.885 + 0.464i)T \)
59 \( 1 + (0.748 + 0.663i)T \)
61 \( 1 + (-0.970 - 0.239i)T \)
67 \( 1 + (0.354 - 0.935i)T \)
71 \( 1 + (-0.885 - 0.464i)T \)
73 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (0.748 - 0.663i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (-0.970 - 0.239i)T \)
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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

−24.49609090253661944998454262574, −23.52514433800316940979329296852, −22.48784684806786795697405255940, −21.99165275107153888979143793188, −20.6748198533527650149065069015, −19.78348419243025036244338108946, −19.00288083916143615684258333892, −17.78677050212688553866062166300, −17.142055362939918825010306029682, −16.16123219198804970498378091271, −15.775556816377917548230430746078, −13.7308378685255109373498438706, −13.35049318334285639610474888798, −12.306144096550356708936374835418, −11.584908397903699837728631043505, −10.33257185256736823782042349215, −9.50657524545735566390000480905, −8.18530938166479950797073873603, −7.1905968231057817554054972147, −6.04281334264352649896600057764, −5.27902139464046850883456241501, −4.13374505277515961892946485086, −2.55096954637610905341210616062, −0.85723884360263695407357910424, −0.18589829506641820416282370213, 1.96266135428672798118239434866, 3.27003378795800997275443503115, 4.43284378317838147601505932302, 5.6387167717827234177341122590, 6.61759588214650795627812014839, 7.23740375213632188866102362515, 9.08482027913848313031289006034, 10.024396182839164505401037598730, 10.531976017145347388881213888911, 11.916145141450854473415981913925, 12.3362041074033349528227365999, 13.76175970237376079513407408094, 14.9120591417540225736372412068, 15.60805866118020988544875517620, 16.46354427576499481139899296160, 17.644408137282259087237277833963, 18.15859169987236488457014901614, 19.22644917644583778032907839403, 20.17468807777941037843426070784, 21.63141563516277012593248662970, 22.01769669393721905553989484696, 22.7673854331652721138912404068, 23.5212367284335622652600131715, 24.67518839398923095197489326804, 25.858319588041307646254290458

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