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Properties

Label 1-315-315.67-r1-0-0
Degree $1$
Conductor $315$
Sign $-0.373 - 0.927i$
Analytic cond. $33.8514$
Root an. cond. $33.8514$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s − i·8-s + 11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)22-s − i·23-s + (−0.5 + 0.866i)26-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + (−0.866 − 0.5i)32-s + (0.5 − 0.866i)34-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s − i·8-s + 11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)22-s − i·23-s + (−0.5 + 0.866i)26-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + (−0.866 − 0.5i)32-s + (0.5 − 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.373 - 0.927i$
Analytic conductor: \(33.8514\)
Root analytic conductor: \(33.8514\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 315,\ (1:\ ),\ -0.373 - 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.819102055 - 2.692551230i\)
\(L(\frac12)\) \(\approx\) \(1.819102055 - 2.692551230i\)
\(L(1)\) \(\approx\) \(1.590643153 - 0.8817921051i\)
\(L(1)\) \(\approx\) \(1.590643153 - 0.8817921051i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−25.20324604865640825712365467067, −24.29929107180322034768429647644, −23.52213355779290605973879991675, −22.47792935886956250457273416146, −21.99315353485899398404509899023, −20.9398009428824786829881466271, −20.05990606297916650640502731084, −19.11322782212749846411176291439, −17.69490946675731518560929085552, −16.96296219223956706785481342926, −16.15019187578840017327349100666, −14.97811223128100540839516681055, −14.44142241224485678173518210881, −13.459787455817509851987700454990, −12.30895860526444996780864126186, −11.8382336096395855876006047880, −10.48088046562130520255305373098, −9.275720898823529679281243482611, −7.98786965874135745399043968909, −7.21261633310184971397890709942, −6.0392250992980890221457276270, −5.20608343718433893514237484267, −3.9602710079166850638504726434, −3.07369340362078897118979668139, −1.535632404015068363976714029122, 0.72172027505975905428046262468, 2.088891007328345546607742690313, 3.22919026390976162497056318969, 4.37356257063319968284411301151, 5.27016054519983292942238542821, 6.51065266628338894043251698392, 7.32709481335156000771061586152, 9.03248517752537663042162974506, 9.87413374357719976909029077320, 10.97625357244767800994716314604, 11.98727061771725532612836630550, 12.48569865150328039915998641566, 13.95058432655886061329697471067, 14.28336691047990275607954263592, 15.40567440409616996280164030565, 16.38823287313805696940093218432, 17.412689436249878554324759682182, 18.74183297568167596385964674648, 19.48244238684082274742207623846, 20.285240388564086578462122051988, 21.23556178430493122159352782294, 22.09807586945192498775556029025, 22.709750333817848782101429789962, 23.74453429326017269703240062, 24.57568371242803581367417035802

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