Properties

Label 2-42e2-9.7-c1-0-7
Degree 22
Conductor 17641764
Sign 0.5520.833i0.552 - 0.833i
Analytic cond. 14.085614.0856
Root an. cond. 3.753083.75308
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.69 − 0.353i)3-s + (0.469 + 0.812i)5-s + (2.75 + 1.19i)9-s + (−1.31 + 2.28i)11-s + (−2.71 − 4.69i)13-s + (−0.508 − 1.54i)15-s + 3.85·17-s − 1.09·19-s + (3.16 + 5.48i)23-s + (2.05 − 3.56i)25-s + (−4.24 − 3.00i)27-s + (−1.94 + 3.36i)29-s + (2.33 + 4.04i)31-s + (3.04 − 3.40i)33-s + 2.30·37-s + ⋯
L(s)  = 1  + (−0.979 − 0.203i)3-s + (0.209 + 0.363i)5-s + (0.916 + 0.399i)9-s + (−0.397 + 0.688i)11-s + (−0.751 − 1.30i)13-s + (−0.131 − 0.398i)15-s + 0.934·17-s − 0.251·19-s + (0.660 + 1.14i)23-s + (0.411 − 0.713i)25-s + (−0.816 − 0.577i)27-s + (−0.360 + 0.624i)29-s + (0.419 + 0.725i)31-s + (0.529 − 0.592i)33-s + 0.379·37-s + ⋯

Functional equation

Λ(s)=(1764s/2ΓC(s)L(s)=((0.5520.833i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1764s/2ΓC(s+1/2)L(s)=((0.5520.833i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    2232722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: 0.5520.833i0.552 - 0.833i
Analytic conductor: 14.085614.0856
Root analytic conductor: 3.753083.75308
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1764(1177,)\chi_{1764} (1177, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :1/2), 0.5520.833i)(2,\ 1764,\ (\ :1/2),\ 0.552 - 0.833i)

Particular Values

L(1)L(1) \approx 1.0507909751.050790975
L(12)L(\frac12) \approx 1.0507909751.050790975
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(1.69+0.353i)T 1 + (1.69 + 0.353i)T
7 1 1
good5 1+(0.4690.812i)T+(2.5+4.33i)T2 1 + (-0.469 - 0.812i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.312.28i)T+(5.59.52i)T2 1 + (1.31 - 2.28i)T + (-5.5 - 9.52i)T^{2}
13 1+(2.71+4.69i)T+(6.5+11.2i)T2 1 + (2.71 + 4.69i)T + (-6.5 + 11.2i)T^{2}
17 13.85T+17T2 1 - 3.85T + 17T^{2}
19 1+1.09T+19T2 1 + 1.09T + 19T^{2}
23 1+(3.165.48i)T+(11.5+19.9i)T2 1 + (-3.16 - 5.48i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.943.36i)T+(14.525.1i)T2 1 + (1.94 - 3.36i)T + (-14.5 - 25.1i)T^{2}
31 1+(2.334.04i)T+(15.5+26.8i)T2 1 + (-2.33 - 4.04i)T + (-15.5 + 26.8i)T^{2}
37 12.30T+37T2 1 - 2.30T + 37T^{2}
41 1+(4.12+7.13i)T+(20.5+35.5i)T2 1 + (4.12 + 7.13i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.143.71i)T+(21.537.2i)T2 1 + (2.14 - 3.71i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.32+2.28i)T+(23.540.7i)T2 1 + (-1.32 + 2.28i)T + (-23.5 - 40.7i)T^{2}
53 1+1.27T+53T2 1 + 1.27T + 53T^{2}
59 1+(3.025.23i)T+(29.5+51.0i)T2 1 + (-3.02 - 5.23i)T + (-29.5 + 51.0i)T^{2}
61 1+(6.7111.6i)T+(30.552.8i)T2 1 + (6.71 - 11.6i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.646.31i)T+(33.5+58.0i)T2 1 + (-3.64 - 6.31i)T + (-33.5 + 58.0i)T^{2}
71 114.7T+71T2 1 - 14.7T + 71T^{2}
73 1+5.74T+73T2 1 + 5.74T + 73T^{2}
79 1+(5.519.54i)T+(39.568.4i)T2 1 + (5.51 - 9.54i)T + (-39.5 - 68.4i)T^{2}
83 1+(1.242.15i)T+(41.571.8i)T2 1 + (1.24 - 2.15i)T + (-41.5 - 71.8i)T^{2}
89 113.5T+89T2 1 - 13.5T + 89T^{2}
97 1+(1.75+3.03i)T+(48.584.0i)T2 1 + (-1.75 + 3.03i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.811519132497296051682152415322, −8.549960275701370528970562850931, −7.46267648757675554983723052898, −7.22012646043601962713198896625, −6.12155861171008315406860100431, −5.33000211200466058999190658494, −4.83492789244022349656478981932, −3.47090339652815279905726140302, −2.38997450012751108376288751538, −1.03223110847777616672638492261, 0.54825955076921349334448080206, 1.89830263167812687633399299272, 3.29662638909496176090765054257, 4.50540588808464240856194354470, 5.00526120864087576722875937199, 5.96017072413000915988877392987, 6.61398850633789489292771503573, 7.48695886743064120524452440667, 8.427066630213209039497481256198, 9.413597974143889936702050471757

Graph of the ZZ-function along the critical line