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Properties

Label 1-312-312.269-r1-0-0
Degree $1$
Conductor $312$
Sign $0.964 - 0.265i$
Analytic cond. $33.5290$
Root an. cond. $33.5290$
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  + 5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + 25-s + (−0.5 − 0.866i)29-s + 31-s + (−0.5 + 0.866i)35-s + (0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s − 47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + 5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + 25-s + (−0.5 − 0.866i)29-s + 31-s + (−0.5 + 0.866i)35-s + (0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s − 47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(33.5290\)
Root analytic conductor: \(33.5290\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 312,\ (1:\ ),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.219530381 - 0.2994487650i\)
\(L(\frac12)\) \(\approx\) \(2.219530381 - 0.2994487650i\)
\(L(1)\) \(\approx\) \(1.282128031 - 0.03289040035i\)
\(L(1)\) \(\approx\) \(1.282128031 - 0.03289040035i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−25.15080221174969362575514811658, −24.21702240871833317079628026135, −23.06073477176455980380167655160, −22.570954299322280664109054664013, −21.29183372210265952127587357814, −20.69526031972384965483486006843, −19.76392391021742829295664736036, −18.6448248173498499019361684620, −17.7519279350016425515991877317, −16.92327658352380753526299288518, −16.17236789747366612570135388494, −14.80230985152613486165284384569, −14.07906404400611956167234170961, −12.996099243691842937694654427173, −12.46999600340320984840896253036, −10.77977050056291243379163187033, −10.14463922118311913824580050062, −9.36167741667403322715969773342, −7.97583097212008596672268849522, −6.93462226420847784289723001584, −5.97256636762939587918481606101, −4.85434412956693063684910322394, −3.59560805377869176887591539155, −2.28211426833200650626337588806, −1.03210998100462291653339601133, 0.82019068857016229462907972727, 2.42984789122909523537898114360, 3.167233953130792849210948759120, 5.044314447245213161460938943506, 5.74547681564961332263521919268, 6.71721190351518430745142900013, 8.0806778015351116574378574459, 9.25944391383033771235229094672, 9.77204535778447846707350508512, 11.07839838779587240203505983649, 12.01621387998681008198903969085, 13.30430293411993906066473714242, 13.66469727493765526596090202036, 14.9981387249281524523714454050, 15.89727876653529630415584795524, 16.79232131278504827137844914073, 17.864638565570353653984858407813, 18.600157481853458774007155814571, 19.419142242646436462155988879782, 20.74274168079906144938298067389, 21.44158279587228840020603062745, 22.145386308683433534697729923922, 23.0679130200795859765771583158, 24.34467202146730336742365993675, 24.94364409332480860854938268745

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