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Properties

Label 2-1350-9.4-c1-0-15
Degree 22
Conductor 13501350
Sign −0.939−0.342i-0.939 - 0.342i
Analytic cond. 10.779810.7798
Root an. cond. 3.283263.28326
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1 + 1.73i)7-s + 0.999·8-s + (−2 + 3.46i)13-s + (0.999 − 1.73i)14-s + (−0.5 − 0.866i)16-s − 6·17-s − 7·19-s + 3.99·26-s − 1.99·28-s + (−3 − 5.19i)29-s + (5 − 8.66i)31-s + (−0.499 + 0.866i)32-s + (3 + 5.19i)34-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.377 + 0.654i)7-s + 0.353·8-s + (−0.554 + 0.960i)13-s + (0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s − 1.45·17-s − 1.60·19-s + 0.784·26-s − 0.377·28-s + (−0.557 − 0.964i)29-s + (0.898 − 1.55i)31-s + (−0.0883 + 0.153i)32-s + (0.514 + 0.891i)34-s + ⋯

Functional equation

Λ(s)=(1350s/2 ΓC(s) L(s)=((−0.939−0.342i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1350s/2 ΓC(s+1/2) L(s)=((−0.939−0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13501350    =    2⋅33⋅522 \cdot 3^{3} \cdot 5^{2}
Sign: −0.939−0.342i-0.939 - 0.342i
Analytic conductor: 10.779810.7798
Root analytic conductor: 3.283263.28326
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1350(901,⋅)\chi_{1350} (901, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 1350, ( :1/2), −0.939−0.342i)(2,\ 1350,\ (\ :1/2),\ -0.939 - 0.342i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1 1
5 1 1
good7 1+(−1−1.73i)T+(−3.5+6.06i)T2 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2}
11 1+(−5.5+9.52i)T2 1 + (-5.5 + 9.52i)T^{2}
13 1+(2−3.46i)T+(−6.5−11.2i)T2 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2}
17 1+6T+17T2 1 + 6T + 17T^{2}
19 1+7T+19T2 1 + 7T + 19T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+(3+5.19i)T+(−14.5+25.1i)T2 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2}
31 1+(−5+8.66i)T+(−15.5−26.8i)T2 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1+(−4.5+7.79i)T+(−20.5−35.5i)T2 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2}
43 1+(0.5+0.866i)T+(−21.5+37.2i)T2 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2}
47 1+(3+5.19i)T+(−23.5+40.7i)T2 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2}
53 1+12T+53T2 1 + 12T + 53T^{2}
59 1+(4.5−7.79i)T+(−29.5−51.0i)T2 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2−3.46i)T+(−30.5+52.8i)T2 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2}
67 1+(6.5−11.2i)T+(−33.5−58.0i)T2 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1−T+73T2 1 - T + 73T^{2}
79 1+(1+1.73i)T+(−39.5+68.4i)T2 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2}
83 1+(4.5+7.79i)T+(−41.5+71.8i)T2 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2}
89 1+15T+89T2 1 + 15T + 89T^{2}
97 1+(−8.5−14.7i)T+(−48.5+84.0i)T2 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−9.037593995785764802196392734289, −8.671884453741680039544031929574, −7.70251703631663084625827794272, −6.72601532554536440988638821294, −5.88242330502159593046046638834, −4.56841762651614364130230098575, −4.10255096288302908897523636584, −2.44540202341530139031734496939, −2.00350616689028588710931413556, 0, 1.60962770375240817886116176686, 3.00594619940090116148398393944, 4.43422675380092256601168355479, 4.90189932926467590833400999713, 6.18403561968746032644569003753, 6.78808972436785101388940283971, 7.68442796592499615592652008523, 8.363251470296773428563682808748, 9.079968569024001693247739654259

Graph of the ZZ-function along the critical line