L(s) = 1 | + (0.866 − 0.5i)2-s + (1.21 − 1.23i)3-s + (0.499 − 0.866i)4-s + (−2.73 − 1.57i)5-s + (0.434 − 1.67i)6-s + (−1.36 + 0.787i)7-s − 0.999i·8-s + (−0.0487 − 2.99i)9-s − 3.15·10-s + (3.26 − 1.88i)11-s + (−0.461 − 1.66i)12-s + (0.899 + 3.49i)13-s + (−0.787 + 1.36i)14-s + (−5.27 + 1.45i)15-s + (−0.5 − 0.866i)16-s + 7.06·17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.701 − 0.712i)3-s + (0.249 − 0.433i)4-s + (−1.22 − 0.706i)5-s + (0.177 − 0.684i)6-s + (−0.515 + 0.297i)7-s − 0.353i·8-s + (−0.0162 − 0.999i)9-s − 0.998·10-s + (0.983 − 0.567i)11-s + (−0.133 − 0.481i)12-s + (0.249 + 0.968i)13-s + (−0.210 + 0.364i)14-s + (−1.36 + 0.376i)15-s + (−0.125 − 0.216i)16-s + 1.71·17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(−0.0806+0.996i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(−0.0806+0.996i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
−0.0806+0.996i
|
Analytic conductor: |
1.86849 |
Root analytic conductor: |
1.36693 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ234(103,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 234, ( :1/2), −0.0806+0.996i)
|
Particular Values
L(1) |
≈ |
1.18683−1.28670i |
L(21) |
≈ |
1.18683−1.28670i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.866+0.5i)T |
| 3 | 1+(−1.21+1.23i)T |
| 13 | 1+(−0.899−3.49i)T |
good | 5 | 1+(2.73+1.57i)T+(2.5+4.33i)T2 |
| 7 | 1+(1.36−0.787i)T+(3.5−6.06i)T2 |
| 11 | 1+(−3.26+1.88i)T+(5.5−9.52i)T2 |
| 17 | 1−7.06T+17T2 |
| 19 | 1−3.76iT−19T2 |
| 23 | 1+(1.84−3.20i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−0.109−0.189i)T+(−14.5+25.1i)T2 |
| 31 | 1+(−2.65−1.53i)T+(15.5+26.8i)T2 |
| 37 | 1−0.292iT−37T2 |
| 41 | 1+(6.39+3.69i)T+(20.5+35.5i)T2 |
| 43 | 1+(3.05+5.29i)T+(−21.5+37.2i)T2 |
| 47 | 1+(−6.17+3.56i)T+(23.5−40.7i)T2 |
| 53 | 1+14.4T+53T2 |
| 59 | 1+(−9.04−5.22i)T+(29.5+51.0i)T2 |
| 61 | 1+(−3.00−5.19i)T+(−30.5+52.8i)T2 |
| 67 | 1+(6.33+3.65i)T+(33.5+58.0i)T2 |
| 71 | 1−0.772iT−71T2 |
| 73 | 1−13.5iT−73T2 |
| 79 | 1+(−6.34−10.9i)T+(−39.5+68.4i)T2 |
| 83 | 1+(0.314−0.181i)T+(41.5−71.8i)T2 |
| 89 | 1+7.06iT−89T2 |
| 97 | 1+(0.535−0.309i)T+(48.5−84.0i)T2 |
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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
−12.02145123408767951084345167380, −11.62300648205128563289799118974, −9.870630339865191437620442281908, −8.858525895562084723612416192837, −8.016544774501680428610182147196, −6.89670210987759316544815688795, −5.74444409714624092867622211688, −3.99428765228656252818037612548, −3.37269240707416043148735516237, −1.34795659634435179667061789313,
3.09207177273390438861760588219, 3.69978281510907938643427472525, 4.82240093081748244250191274566, 6.44788792239047156685714344617, 7.55149038646452515842131844957, 8.192040129866569188071760344769, 9.603458842236166210586608560164, 10.51270363051937595105065219152, 11.53976661376087895129087495304, 12.44627114308290379150557737650