L(s) = 1 | + (0.397 − 1.35i)2-s + (−0.556 + 0.963i)3-s + (−1.68 − 1.07i)4-s + (1.08 + 1.13i)6-s + (−2.32 − 1.26i)7-s + (−2.13 + 1.85i)8-s + (0.880 + 1.52i)9-s + (1.48 + 0.856i)11-s + (1.97 − 1.02i)12-s + 2.45i·13-s + (−2.63 + 2.65i)14-s + (1.67 + 3.63i)16-s + (5.38 + 3.10i)17-s + (2.42 − 0.589i)18-s + (−0.108 − 0.187i)19-s + ⋯ |
L(s) = 1 | + (0.280 − 0.959i)2-s + (−0.321 + 0.556i)3-s + (−0.842 − 0.539i)4-s + (0.443 + 0.464i)6-s + (−0.878 − 0.477i)7-s + (−0.753 + 0.656i)8-s + (0.293 + 0.508i)9-s + (0.447 + 0.258i)11-s + (0.570 − 0.295i)12-s + 0.682i·13-s + (−0.705 + 0.708i)14-s + (0.418 + 0.908i)16-s + (1.30 + 0.754i)17-s + (0.570 − 0.138i)18-s + (−0.0248 − 0.0430i)19-s + ⋯ |
Λ(s)=(=(700s/2ΓC(s)L(s)(0.994−0.106i)Λ(2−s)
Λ(s)=(=(700s/2ΓC(s+1/2)L(s)(0.994−0.106i)Λ(1−s)
Degree: |
2 |
Conductor: |
700
= 22⋅52⋅7
|
Sign: |
0.994−0.106i
|
Analytic conductor: |
5.58952 |
Root analytic conductor: |
2.36421 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ700(451,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 700, ( :1/2), 0.994−0.106i)
|
Particular Values
L(1) |
≈ |
1.23719+0.0658172i |
L(21) |
≈ |
1.23719+0.0658172i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.397+1.35i)T |
| 5 | 1 |
| 7 | 1+(2.32+1.26i)T |
good | 3 | 1+(0.556−0.963i)T+(−1.5−2.59i)T2 |
| 11 | 1+(−1.48−0.856i)T+(5.5+9.52i)T2 |
| 13 | 1−2.45iT−13T2 |
| 17 | 1+(−5.38−3.10i)T+(8.5+14.7i)T2 |
| 19 | 1+(0.108+0.187i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−5.68+3.28i)T+(11.5−19.9i)T2 |
| 29 | 1+2.47T+29T2 |
| 31 | 1+(−0.0819+0.141i)T+(−15.5−26.8i)T2 |
| 37 | 1+(−3.84−6.66i)T+(−18.5+32.0i)T2 |
| 41 | 1−8.34iT−41T2 |
| 43 | 1−1.89iT−43T2 |
| 47 | 1+(−5.85−10.1i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−6.51+11.2i)T+(−26.5−45.8i)T2 |
| 59 | 1+(2.14−3.71i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−6.06+3.50i)T+(30.5−52.8i)T2 |
| 67 | 1+(4.48+2.58i)T+(33.5+58.0i)T2 |
| 71 | 1+5.04iT−71T2 |
| 73 | 1+(6.59+3.80i)T+(36.5+63.2i)T2 |
| 79 | 1+(13.8−7.97i)T+(39.5−68.4i)T2 |
| 83 | 1+5.47T+83T2 |
| 89 | 1+(1.54−0.891i)T+(44.5−77.0i)T2 |
| 97 | 1−10.5iT−97T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.44969795530818830857067218973, −9.827945193361549710365836697786, −9.227844993903476010662717693969, −8.007892511963995950851176805699, −6.76486549179248769710930611300, −5.76964824487436583532938571807, −4.66898539486003321998415163607, −3.97234369795929099900781732473, −2.91539227199489866923311377884, −1.32693137140157469890987492795,
0.72399192438142079271456235658, 3.03585424602657936768203383104, 3.92226642317027092728086596580, 5.60817160967029535757278266907, 5.75781628503234341141731081788, 7.13505431875915364610112162841, 7.29082886882047995230084331770, 8.715749173753059629506882223672, 9.342657686944132480997231064503, 10.17389676350916523495598228241