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Properties

Label 1-465-465.359-r1-0-0
Degree $1$
Conductor $465$
Sign $0.987 - 0.157i$
Analytic cond. $49.9711$
Root an. cond. $49.9711$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.104 − 0.994i)7-s + (−0.809 + 0.587i)8-s + (−0.913 + 0.406i)11-s + (0.978 + 0.207i)13-s + (−0.913 − 0.406i)14-s + (0.309 + 0.951i)16-s + (0.913 + 0.406i)17-s + (−0.978 + 0.207i)19-s + (0.104 + 0.994i)22-s + (−0.809 + 0.587i)23-s + (0.5 − 0.866i)26-s + (−0.669 + 0.743i)28-s + (−0.309 + 0.951i)29-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.104 − 0.994i)7-s + (−0.809 + 0.587i)8-s + (−0.913 + 0.406i)11-s + (0.978 + 0.207i)13-s + (−0.913 − 0.406i)14-s + (0.309 + 0.951i)16-s + (0.913 + 0.406i)17-s + (−0.978 + 0.207i)19-s + (0.104 + 0.994i)22-s + (−0.809 + 0.587i)23-s + (0.5 − 0.866i)26-s + (−0.669 + 0.743i)28-s + (−0.309 + 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.987 - 0.157i$
Analytic conductor: \(49.9711\)
Root analytic conductor: \(49.9711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 465,\ (1:\ ),\ 0.987 - 0.157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.473750057 - 0.1166601759i\)
\(L(\frac12)\) \(\approx\) \(1.473750057 - 0.1166601759i\)
\(L(1)\) \(\approx\) \(0.9507890149 - 0.4528356483i\)
\(L(1)\) \(\approx\) \(0.9507890149 - 0.4528356483i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.104 - 0.994i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (0.978 + 0.207i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.669 - 0.743i)T \)
43 \( 1 + (0.978 - 0.207i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.104 + 0.994i)T \)
73 \( 1 + (-0.913 + 0.406i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−23.61378198299421425059862928299, −23.12640470732528330129625937089, −22.04372093032720580271579947227, −21.3314578499622479979432177282, −20.63989248377361521362587637391, −18.97484438787462196006919785354, −18.47913072310017114426571029218, −17.70647228647395713238302963461, −16.5440470540927644199409166039, −15.86775592329273211216997454661, −15.16483555891115391601653120924, −14.26921461595771451664033267774, −13.300666759214147893111711536221, −12.57508827913007560211657210103, −11.579167510114149434089685346953, −10.34261192238530440367026556392, −9.1547283327987601076875125107, −8.33011664908652501447192100890, −7.6770360727991394809591838612, −6.21124693420189408561807044320, −5.75381377899971129138827988852, −4.69859207656404097669263061518, −3.49658775532267767238507797330, −2.377807711851239621991869105259, −0.409223085469560373173168067059, 1.00897605385513097965345278499, 2.0275395782494086866393109888, 3.41565825659761100460148608493, 4.130785985657852349694180158512, 5.222168942245327120433185110333, 6.27464261761953614951109697371, 7.66013367238922862298711163946, 8.56299864171616767991731776895, 9.83655590629966234492246185570, 10.50848864578562573804984556760, 11.181848615465321526158473216313, 12.36236180451125296623700877562, 13.11130323686229381692432682700, 13.88728837178219389632471868026, 14.707226303146327722952403643542, 15.80882194549479908824669847462, 16.92832953435326249850732536448, 17.85082578624604341122872014833, 18.67583978646153774966773925572, 19.47766542507730111671958485758, 20.533814024701642266682691426722, 20.85836494781777508656933447089, 21.81229068481098776627298065270, 22.86250814555726443879906964015, 23.68136232383966493723087341897

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