Properties

Label 2-700-28.3-c1-0-32
Degree 22
Conductor 700700
Sign 0.506+0.862i0.506 + 0.862i
Analytic cond. 5.589525.58952
Root an. cond. 2.364212.36421
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  + (−0.288 − 1.38i)2-s + (0.450 − 0.780i)3-s + (−1.83 + 0.798i)4-s + (−1.21 − 0.398i)6-s + (2.29 − 1.30i)7-s + (1.63 + 2.30i)8-s + (1.09 + 1.89i)9-s + (3.24 + 1.87i)11-s + (−0.202 + 1.79i)12-s + 2.41i·13-s + (−2.47 − 2.80i)14-s + (2.72 − 2.92i)16-s + (0.505 + 0.291i)17-s + (2.30 − 2.06i)18-s + (3.07 + 5.33i)19-s + ⋯
L(s)  = 1  + (−0.204 − 0.978i)2-s + (0.260 − 0.450i)3-s + (−0.916 + 0.399i)4-s + (−0.494 − 0.162i)6-s + (0.869 − 0.494i)7-s + (0.578 + 0.815i)8-s + (0.364 + 0.631i)9-s + (0.977 + 0.564i)11-s + (−0.0585 + 0.517i)12-s + 0.671i·13-s + (−0.661 − 0.750i)14-s + (0.680 − 0.732i)16-s + (0.122 + 0.0707i)17-s + (0.543 − 0.485i)18-s + (0.706 + 1.22i)19-s + ⋯

Functional equation

Λ(s)=(700s/2ΓC(s)L(s)=((0.506+0.862i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2ΓC(s+1/2)L(s)=((0.506+0.862i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    225272^{2} \cdot 5^{2} \cdot 7
Sign: 0.506+0.862i0.506 + 0.862i
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ700(451,)\chi_{700} (451, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), 0.506+0.862i)(2,\ 700,\ (\ :1/2),\ 0.506 + 0.862i)

Particular Values

L(1)L(1) \approx 1.403830.803268i1.40383 - 0.803268i
L(12)L(\frac12) \approx 1.403830.803268i1.40383 - 0.803268i
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+(0.288+1.38i)T 1 + (0.288 + 1.38i)T
5 1 1
7 1+(2.29+1.30i)T 1 + (-2.29 + 1.30i)T
good3 1+(0.450+0.780i)T+(1.52.59i)T2 1 + (-0.450 + 0.780i)T + (-1.5 - 2.59i)T^{2}
11 1+(3.241.87i)T+(5.5+9.52i)T2 1 + (-3.24 - 1.87i)T + (5.5 + 9.52i)T^{2}
13 12.41iT13T2 1 - 2.41iT - 13T^{2}
17 1+(0.5050.291i)T+(8.5+14.7i)T2 1 + (-0.505 - 0.291i)T + (8.5 + 14.7i)T^{2}
19 1+(3.075.33i)T+(9.5+16.4i)T2 1 + (-3.07 - 5.33i)T + (-9.5 + 16.4i)T^{2}
23 1+(3.732.15i)T+(11.519.9i)T2 1 + (3.73 - 2.15i)T + (11.5 - 19.9i)T^{2}
29 1+0.435T+29T2 1 + 0.435T + 29T^{2}
31 1+(1.26+2.19i)T+(15.526.8i)T2 1 + (-1.26 + 2.19i)T + (-15.5 - 26.8i)T^{2}
37 1+(5.65+9.78i)T+(18.5+32.0i)T2 1 + (5.65 + 9.78i)T + (-18.5 + 32.0i)T^{2}
41 1+7.35iT41T2 1 + 7.35iT - 41T^{2}
43 1+5.80iT43T2 1 + 5.80iT - 43T^{2}
47 1+(5.7810.0i)T+(23.5+40.7i)T2 1 + (-5.78 - 10.0i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.552.69i)T+(26.545.8i)T2 1 + (1.55 - 2.69i)T + (-26.5 - 45.8i)T^{2}
59 1+(1.73+3.00i)T+(29.551.0i)T2 1 + (-1.73 + 3.00i)T + (-29.5 - 51.0i)T^{2}
61 1+(8.995.19i)T+(30.552.8i)T2 1 + (8.99 - 5.19i)T + (30.5 - 52.8i)T^{2}
67 1+(8.524.92i)T+(33.5+58.0i)T2 1 + (-8.52 - 4.92i)T + (33.5 + 58.0i)T^{2}
71 1+9.96iT71T2 1 + 9.96iT - 71T^{2}
73 1+(8.48+4.89i)T+(36.5+63.2i)T2 1 + (8.48 + 4.89i)T + (36.5 + 63.2i)T^{2}
79 1+(0.397+0.229i)T+(39.568.4i)T2 1 + (-0.397 + 0.229i)T + (39.5 - 68.4i)T^{2}
83 1+2.59T+83T2 1 + 2.59T + 83T^{2}
89 1+(8.554.94i)T+(44.577.0i)T2 1 + (8.55 - 4.94i)T + (44.5 - 77.0i)T^{2}
97 14.54iT97T2 1 - 4.54iT - 97T^{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

−10.39295363427268366571405206908, −9.552315658691274286267401492955, −8.685078148088543470546396724485, −7.70458216584139071539264597139, −7.24505093422419116895140097668, −5.61994972933546055765557827627, −4.41969111333366381119231629663, −3.78140005486574857981845541544, −2.06334771878622122453001655337, −1.42747652266068465734193393270, 1.13199219369193030973908726704, 3.21440342933127530425380718191, 4.37439883188040736035252330151, 5.18473240947939256646899729109, 6.23258650267539193696138988422, 7.03984878064138429624432859154, 8.193359014806216574932042231883, 8.724924930497671790525700281345, 9.517320483380932721221831261726, 10.26868032315461018657650697784

Graph of the ZZ-function along the critical line