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Properties

Label 2-1260-105.2-c1-0-5
Degree $2$
Conductor $1260$
Sign $0.272 - 0.962i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.13 + 0.675i)5-s + (2.61 + 0.407i)7-s + (−3.45 + 1.99i)11-s + (−4.61 + 4.61i)13-s + (5.02 + 1.34i)17-s + (4.70 + 2.71i)19-s + (−8.33 + 2.23i)23-s + (4.08 + 2.88i)25-s − 3.86·29-s + (−0.780 − 1.35i)31-s + (5.29 + 2.63i)35-s + (−0.275 + 0.0737i)37-s + 3.13i·41-s + (2.78 − 2.78i)43-s + (0.204 + 0.765i)47-s + ⋯
L(s)  = 1  + (0.953 + 0.302i)5-s + (0.988 + 0.154i)7-s + (−1.04 + 0.601i)11-s + (−1.28 + 1.28i)13-s + (1.21 + 0.326i)17-s + (1.07 + 0.622i)19-s + (−1.73 + 0.465i)23-s + (0.817 + 0.576i)25-s − 0.716·29-s + (−0.140 − 0.242i)31-s + (0.895 + 0.445i)35-s + (−0.0452 + 0.0121i)37-s + 0.490i·41-s + (0.423 − 0.423i)43-s + (0.0299 + 0.111i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.272 - 0.962i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.272 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.867152144\)
\(L(\frac12)\) \(\approx\) \(1.867152144\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2.13 - 0.675i)T \)
7 \( 1 + (-2.61 - 0.407i)T \)
good11 \( 1 + (3.45 - 1.99i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.61 - 4.61i)T - 13iT^{2} \)
17 \( 1 + (-5.02 - 1.34i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-4.70 - 2.71i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.33 - 2.23i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 3.86T + 29T^{2} \)
31 \( 1 + (0.780 + 1.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.275 - 0.0737i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.13iT - 41T^{2} \)
43 \( 1 + (-2.78 + 2.78i)T - 43iT^{2} \)
47 \( 1 + (-0.204 - 0.765i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.60 + 5.99i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.19 + 9.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.58 + 7.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.52 + 5.70i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.668iT - 71T^{2} \)
73 \( 1 + (-10.7 - 2.89i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-13.8 - 8.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.96 + 3.96i)T + 83iT^{2} \)
89 \( 1 + (5.68 - 9.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.4 - 12.4i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.782253920195967469561263732370, −9.381393063942842541332053253395, −7.85767891420656747891336375230, −7.70271509596349926893959888015, −6.53281423032311899794708123017, −5.41176738072059778177628485554, −5.07998388391307630980616071129, −3.77433339036775590768968552125, −2.31503711594184891531946924000, −1.74703334596429197454108325022, 0.77528003978631876072953845956, 2.21967548615077508962835683433, 3.08551535203218765263611281732, 4.63983471697749669750411938494, 5.49763426514781661044000251308, 5.70255377848285835916517916224, 7.39665778061291148569936078529, 7.77917505140695940338092112972, 8.641787358227204240339704769538, 9.737888264342054580676057609481

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