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Properties

Label 2-294-7.2-c5-0-18
Degree 22
Conductor 294294
Sign 0.947+0.318i0.947 + 0.318i
Analytic cond. 47.152847.1528
Root an. cond. 6.866796.86679
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 3.46i)2-s + (4.5 + 7.79i)3-s + (−7.99 − 13.8i)4-s + (−30.5 + 52.8i)5-s − 36·6-s + 63.9·8-s + (−40.5 + 70.1i)9-s + (−122. − 211. i)10-s + (18.2 + 31.6i)11-s + (72 − 124. i)12-s − 34.5·13-s − 549.·15-s + (−128 + 221. i)16-s + (−1.03e3 − 1.78e3i)17-s + (−162 − 280. i)18-s + (226. − 391. i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.546 + 0.946i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.386 − 0.669i)10-s + (0.0455 + 0.0788i)11-s + (0.144 − 0.249i)12-s − 0.0567·13-s − 0.630·15-s + (−0.125 + 0.216i)16-s + (−0.864 − 1.49i)17-s + (−0.117 − 0.204i)18-s + (0.143 − 0.248i)19-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((0.947+0.318i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=((0.947+0.318i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: 0.947+0.318i0.947 + 0.318i
Analytic conductor: 47.152847.1528
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ294(79,⋅)\chi_{294} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 294, ( :5/2), 0.947+0.318i)(2,\ 294,\ (\ :5/2),\ 0.947 + 0.318i)

Particular Values

L(3)L(3) ≈\approx 0.74982073360.7498207336
L(12)L(\frac12) ≈\approx 0.74982073360.7498207336
L(72)L(\frac{7}{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+(2−3.46i)T 1 + (2 - 3.46i)T
3 1+(−4.5−7.79i)T 1 + (-4.5 - 7.79i)T
7 1 1
good5 1+(30.5−52.8i)T+(−1.56e3−2.70e3i)T2 1 + (30.5 - 52.8i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(−18.2−31.6i)T+(−8.05e4+1.39e5i)T2 1 + (-18.2 - 31.6i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1+34.5T+3.71e5T2 1 + 34.5T + 3.71e5T^{2}
17 1+(1.03e3+1.78e3i)T+(−7.09e5+1.22e6i)T2 1 + (1.03e3 + 1.78e3i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−226.+391.i)T+(−1.23e6−2.14e6i)T2 1 + (-226. + 391. i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(842.−1.45e3i)T+(−3.21e6−5.57e6i)T2 1 + (842. - 1.45e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1+4.76e3T+2.05e7T2 1 + 4.76e3T + 2.05e7T^{2}
31 1+(2.63e3+4.55e3i)T+(−1.43e7+2.47e7i)T2 1 + (2.63e3 + 4.55e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(−6.41e3+1.11e4i)T+(−3.46e7−6.00e7i)T2 1 + (-6.41e3 + 1.11e4i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1−7.12e3T+1.15e8T2 1 - 7.12e3T + 1.15e8T^{2}
43 1−1.11e4T+1.47e8T2 1 - 1.11e4T + 1.47e8T^{2}
47 1+(1.17e4−2.03e4i)T+(−1.14e8−1.98e8i)T2 1 + (1.17e4 - 2.03e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−3.51e3−6.08e3i)T+(−2.09e8+3.62e8i)T2 1 + (-3.51e3 - 6.08e3i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(2.21e4+3.82e4i)T+(−3.57e8+6.19e8i)T2 1 + (2.21e4 + 3.82e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−9.69e3+1.67e4i)T+(−4.22e8−7.31e8i)T2 1 + (-9.69e3 + 1.67e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(1.04e4+1.81e4i)T+(−6.75e8+1.16e9i)T2 1 + (1.04e4 + 1.81e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1−7.98e4T+1.80e9T2 1 - 7.98e4T + 1.80e9T^{2}
73 1+(1.85e4+3.20e4i)T+(−1.03e9+1.79e9i)T2 1 + (1.85e4 + 3.20e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(2.10e4−3.64e4i)T+(−1.53e9−2.66e9i)T2 1 + (2.10e4 - 3.64e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−6.31e3T+3.93e9T2 1 - 6.31e3T + 3.93e9T^{2}
89 1+(2.57e4−4.45e4i)T+(−2.79e9−4.83e9i)T2 1 + (2.57e4 - 4.45e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1−1.27e5T+8.58e9T2 1 - 1.27e5T + 8.58e9T^{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

−11.10013684943588723215929444153, −9.618036268444725818201688636104, −9.182925359377721862656260444208, −7.72862656859814840586007141855, −7.26832845584367800222180109021, −6.06255575862572756083903982230, −4.79588065535016770624800005026, −3.63970699014396584381104725703, −2.37213439224294875715512849939, −0.25784404514436090916783576927, 1.00948000079525689390794197276, 2.11931261879410886376081098485, 3.63877585010834601417724967217, 4.59906983269689322257651654206, 6.08281657980994776768704224692, 7.40189167177027544402155400617, 8.429926355187836498322276458573, 8.800595384431472337308056469854, 10.04069961796474880417794398804, 11.07567510895942704842644818402

Graph of the ZZ-function along the critical line