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Properties

Label 2-1260-105.2-c1-0-11
Degree $2$
Conductor $1260$
Sign $0.0776 + 0.996i$
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  + (−2.05 + 0.880i)5-s + (−1.99 + 1.74i)7-s + (−0.448 + 0.259i)11-s + (2.98 − 2.98i)13-s + (−2.86 − 0.768i)17-s + (−1.69 − 0.979i)19-s + (0.852 − 0.228i)23-s + (3.44 − 3.62i)25-s − 5.32·29-s + (0.959 + 1.66i)31-s + (2.55 − 5.33i)35-s + (5.11 − 1.37i)37-s − 9.17i·41-s + (8.40 − 8.40i)43-s + (0.865 + 3.23i)47-s + ⋯
L(s)  = 1  + (−0.919 + 0.393i)5-s + (−0.752 + 0.658i)7-s + (−0.135 + 0.0781i)11-s + (0.828 − 0.828i)13-s + (−0.695 − 0.186i)17-s + (−0.389 − 0.224i)19-s + (0.177 − 0.0476i)23-s + (0.689 − 0.724i)25-s − 0.988·29-s + (0.172 + 0.298i)31-s + (0.432 − 0.901i)35-s + (0.841 − 0.225i)37-s − 1.43i·41-s + (1.28 − 1.28i)43-s + (0.126 + 0.471i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0776 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0776 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0776 + 0.996i$
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: \((2,\ 1260,\ (\ :1/2),\ 0.0776 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7096302794\)
\(L(\frac12)\) \(\approx\) \(0.7096302794\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 - 0.880i)T \)
7 \( 1 + (1.99 - 1.74i)T \)
good11 \( 1 + (0.448 - 0.259i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.98 + 2.98i)T - 13iT^{2} \)
17 \( 1 + (2.86 + 0.768i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.69 + 0.979i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.852 + 0.228i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.32T + 29T^{2} \)
31 \( 1 + (-0.959 - 1.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.11 + 1.37i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 9.17iT - 41T^{2} \)
43 \( 1 + (-8.40 + 8.40i)T - 43iT^{2} \)
47 \( 1 + (-0.865 - 3.23i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.945 + 3.52i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.01 + 6.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.45 + 4.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.262 + 0.978i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.99iT - 71T^{2} \)
73 \( 1 + (4.11 + 1.10i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.16 + 3.55i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.26 - 2.26i)T + 83iT^{2} \)
89 \( 1 + (-0.108 + 0.187i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.17 + 6.17i)T + 97iT^{2} \)
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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

−9.329072586703543770923546907678, −8.725793105633057963429099886034, −7.87269984605102034190029564758, −7.03315328508928502152875688813, −6.22088380878738654016144654196, −5.35558081962110757212245667692, −4.11958803893285536619452770748, −3.32293424213584845279615117055, −2.35467469479861569068323199920, −0.33320757095913897063389831821, 1.19883756541555648681903457638, 2.87315974311063443805184809188, 4.04943497804936528867106410851, 4.35903420535185011575120681640, 5.83093317736410092392436997254, 6.64108070028841678136831704703, 7.44202970834200622657077214602, 8.259627233973186512470356912567, 9.059783649753618023589130714471, 9.758966221412572863774476539446

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