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Properties

Label 2-702-117.103-c1-0-8
Degree 22
Conductor 702702
Sign 0.997−0.0673i0.997 - 0.0673i
Analytic cond. 5.605495.60549
Root an. cond. 2.367592.36759
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (2.53 + 1.46i)5-s + (1.90 − 1.10i)7-s + 0.999i·8-s − 2.93·10-s + (4.47 − 2.58i)11-s + (0.680 − 3.54i)13-s + (−1.10 + 1.90i)14-s + (−0.5 − 0.866i)16-s − 2.31·17-s − 5.16i·19-s + (2.53 − 1.46i)20-s + (−2.58 + 4.47i)22-s + (−4.19 + 7.26i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (1.13 + 0.655i)5-s + (0.721 − 0.416i)7-s + 0.353i·8-s − 0.927·10-s + (1.34 − 0.779i)11-s + (0.188 − 0.982i)13-s + (−0.294 + 0.510i)14-s + (−0.125 − 0.216i)16-s − 0.560·17-s − 1.18i·19-s + (0.567 − 0.327i)20-s + (−0.551 + 0.954i)22-s + (−0.874 + 1.51i)23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((0.997−0.0673i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0673i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((0.997−0.0673i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0673i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: 0.997−0.0673i0.997 - 0.0673i
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ702(415,⋅)\chi_{702} (415, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 702, ( :1/2), 0.997−0.0673i)(2,\ 702,\ (\ :1/2),\ 0.997 - 0.0673i)

Particular Values

L(1)L(1) ≈\approx 1.55430+0.0524286i1.55430 + 0.0524286i
L(12)L(\frac12) ≈\approx 1.55430+0.0524286i1.55430 + 0.0524286i
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+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
3 1 1
13 1+(−0.680+3.54i)T 1 + (-0.680 + 3.54i)T
good5 1+(−2.53−1.46i)T+(2.5+4.33i)T2 1 + (-2.53 - 1.46i)T + (2.5 + 4.33i)T^{2}
7 1+(−1.90+1.10i)T+(3.5−6.06i)T2 1 + (-1.90 + 1.10i)T + (3.5 - 6.06i)T^{2}
11 1+(−4.47+2.58i)T+(5.5−9.52i)T2 1 + (-4.47 + 2.58i)T + (5.5 - 9.52i)T^{2}
17 1+2.31T+17T2 1 + 2.31T + 17T^{2}
19 1+5.16iT−19T2 1 + 5.16iT - 19T^{2}
23 1+(4.19−7.26i)T+(−11.5−19.9i)T2 1 + (4.19 - 7.26i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.72+8.18i)T+(−14.5+25.1i)T2 1 + (4.72 + 8.18i)T + (-14.5 + 25.1i)T^{2}
31 1+(−5.38−3.11i)T+(15.5+26.8i)T2 1 + (-5.38 - 3.11i)T + (15.5 + 26.8i)T^{2}
37 1−0.646iT−37T2 1 - 0.646iT - 37T^{2}
41 1+(0.674+0.389i)T+(20.5+35.5i)T2 1 + (0.674 + 0.389i)T + (20.5 + 35.5i)T^{2}
43 1+(1.74+3.02i)T+(−21.5+37.2i)T2 1 + (1.74 + 3.02i)T + (-21.5 + 37.2i)T^{2}
47 1+(4.79−2.76i)T+(23.5−40.7i)T2 1 + (4.79 - 2.76i)T + (23.5 - 40.7i)T^{2}
53 1−8.68T+53T2 1 - 8.68T + 53T^{2}
59 1+(−2.59−1.49i)T+(29.5+51.0i)T2 1 + (-2.59 - 1.49i)T + (29.5 + 51.0i)T^{2}
61 1+(−0.432−0.748i)T+(−30.5+52.8i)T2 1 + (-0.432 - 0.748i)T + (-30.5 + 52.8i)T^{2}
67 1+(−9.68−5.58i)T+(33.5+58.0i)T2 1 + (-9.68 - 5.58i)T + (33.5 + 58.0i)T^{2}
71 1−12.9iT−71T2 1 - 12.9iT - 71T^{2}
73 1−4.27iT−73T2 1 - 4.27iT - 73T^{2}
79 1+(−1.52−2.63i)T+(−39.5+68.4i)T2 1 + (-1.52 - 2.63i)T + (-39.5 + 68.4i)T^{2}
83 1+(3.19−1.84i)T+(41.5−71.8i)T2 1 + (3.19 - 1.84i)T + (41.5 - 71.8i)T^{2}
89 1+3.34iT−89T2 1 + 3.34iT - 89T^{2}
97 1+(−1.29+0.745i)T+(48.5−84.0i)T2 1 + (-1.29 + 0.745i)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

−10.25984156281714405233784983164, −9.645283445047116281014597892240, −8.773351123218174604043928957730, −7.892749799027155314944085049108, −6.87592862132687420124617124055, −6.14747640731431856750092873503, −5.36159742671088979106186781352, −3.89112073650897991948232031280, −2.45254401083262215578841471986, −1.17704384331646912915202843204, 1.59106626614899714877848119821, 2.03637927466372211988026117283, 3.97030661419768999517583371419, 4.89811689205850065609341816916, 6.13780528528259468443349241371, 6.83057075489685691830519796196, 8.207068916023249583444840723750, 8.881403753247296241350480053546, 9.509231372288929850338608872828, 10.20430439583584331645325359765

Graph of the ZZ-function along the critical line