Properties

Label 2-850-85.78-c1-0-18
Degree 22
Conductor 850850
Sign 0.767+0.641i0.767 + 0.641i
Analytic cond. 6.787286.78728
Root an. cond. 2.605242.60524
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  + (0.382 + 0.923i)2-s + (−0.442 − 2.22i)3-s + (−0.707 + 0.707i)4-s + (1.88 − 1.26i)6-s + (1.93 − 1.29i)7-s + (−0.923 − 0.382i)8-s + (−1.98 + 0.820i)9-s + (1.86 + 2.78i)11-s + (1.88 + 1.26i)12-s + 4.73·13-s + (1.93 + 1.29i)14-s i·16-s + (−3.36 − 2.38i)17-s + (−1.51 − 1.51i)18-s + (0.786 + 0.325i)19-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (−0.255 − 1.28i)3-s + (−0.353 + 0.353i)4-s + (0.769 − 0.514i)6-s + (0.731 − 0.488i)7-s + (−0.326 − 0.135i)8-s + (−0.660 + 0.273i)9-s + (0.561 + 0.840i)11-s + (0.544 + 0.363i)12-s + 1.31·13-s + (0.517 + 0.345i)14-s − 0.250i·16-s + (−0.815 − 0.578i)17-s + (−0.357 − 0.357i)18-s + (0.180 + 0.0747i)19-s + ⋯

Functional equation

Λ(s)=(850s/2ΓC(s)L(s)=((0.767+0.641i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(850s/2ΓC(s+1/2)L(s)=((0.767+0.641i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 850850    =    252172 \cdot 5^{2} \cdot 17
Sign: 0.767+0.641i0.767 + 0.641i
Analytic conductor: 6.787286.78728
Root analytic conductor: 2.605242.60524
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ850(843,)\chi_{850} (843, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 850, ( :1/2), 0.767+0.641i)(2,\ 850,\ (\ :1/2),\ 0.767 + 0.641i)

Particular Values

L(1)L(1) \approx 1.678240.608860i1.67824 - 0.608860i
L(12)L(\frac12) \approx 1.678240.608860i1.67824 - 0.608860i
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+(0.3820.923i)T 1 + (-0.382 - 0.923i)T
5 1 1
17 1+(3.36+2.38i)T 1 + (3.36 + 2.38i)T
good3 1+(0.442+2.22i)T+(2.77+1.14i)T2 1 + (0.442 + 2.22i)T + (-2.77 + 1.14i)T^{2}
7 1+(1.93+1.29i)T+(2.676.46i)T2 1 + (-1.93 + 1.29i)T + (2.67 - 6.46i)T^{2}
11 1+(1.862.78i)T+(4.20+10.1i)T2 1 + (-1.86 - 2.78i)T + (-4.20 + 10.1i)T^{2}
13 14.73T+13T2 1 - 4.73T + 13T^{2}
19 1+(0.7860.325i)T+(13.4+13.4i)T2 1 + (-0.786 - 0.325i)T + (13.4 + 13.4i)T^{2}
23 1+(4.12+0.820i)T+(21.2+8.80i)T2 1 + (4.12 + 0.820i)T + (21.2 + 8.80i)T^{2}
29 1+(3.83+0.762i)T+(26.711.0i)T2 1 + (-3.83 + 0.762i)T + (26.7 - 11.0i)T^{2}
31 1+(4.45+6.66i)T+(11.828.6i)T2 1 + (-4.45 + 6.66i)T + (-11.8 - 28.6i)T^{2}
37 1+(9.93+1.97i)T+(34.114.1i)T2 1 + (-9.93 + 1.97i)T + (34.1 - 14.1i)T^{2}
41 1+(0.472+0.0940i)T+(37.8+15.6i)T2 1 + (0.472 + 0.0940i)T + (37.8 + 15.6i)T^{2}
43 1+(0.7021.69i)T+(30.430.4i)T2 1 + (0.702 - 1.69i)T + (-30.4 - 30.4i)T^{2}
47 1+6.52iT47T2 1 + 6.52iT - 47T^{2}
53 1+(3.221.33i)T+(37.437.4i)T2 1 + (3.22 - 1.33i)T + (37.4 - 37.4i)T^{2}
59 1+(5.52+13.3i)T+(41.7+41.7i)T2 1 + (5.52 + 13.3i)T + (-41.7 + 41.7i)T^{2}
61 1+(1.175.90i)T+(56.323.3i)T2 1 + (1.17 - 5.90i)T + (-56.3 - 23.3i)T^{2}
67 1+(0.9120.912i)T+67iT2 1 + (-0.912 - 0.912i)T + 67iT^{2}
71 1+(10.97.33i)T+(27.1+65.5i)T2 1 + (-10.9 - 7.33i)T + (27.1 + 65.5i)T^{2}
73 1+(10.1+6.77i)T+(27.9+67.4i)T2 1 + (10.1 + 6.77i)T + (27.9 + 67.4i)T^{2}
79 1+(6.77+4.52i)T+(30.272.9i)T2 1 + (-6.77 + 4.52i)T + (30.2 - 72.9i)T^{2}
83 1+(3.72+8.98i)T+(58.6+58.6i)T2 1 + (3.72 + 8.98i)T + (-58.6 + 58.6i)T^{2}
89 1+(2.752.75i)T89iT2 1 + (2.75 - 2.75i)T - 89iT^{2}
97 1+(7.645.10i)T+(37.1+89.6i)T2 1 + (-7.64 - 5.10i)T + (37.1 + 89.6i)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

−10.03197019496169168831845041946, −8.978908719075742508574841460883, −7.954566712931030284724226822414, −7.57548082999713028868038222777, −6.46903305393840917647153188350, −6.20806897414937869580701338355, −4.76045124175921427944932153392, −3.97416595797263946294554743314, −2.20099098000632130502861244690, −0.977764236950555286324649862660, 1.43670839602344374373775648483, 3.04267092922784627763575169652, 4.02353850746911936133837885265, 4.65633228124694624887690418604, 5.70414260795856169417207432257, 6.39657759614317088863934157769, 8.234218807928529247506070614454, 8.735599649689829582329450107801, 9.580161632461479076186701689807, 10.47886307710487813061876860880

Graph of the ZZ-function along the critical line