L(s) = 1 | + (0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 − 0.587i)4-s + (0.587 + 0.809i)5-s + (0.891 + 0.453i)6-s + (0.891 − 0.453i)7-s + (0.587 − 0.809i)8-s + i·9-s + (0.809 + 0.587i)10-s + (0.156 + 0.987i)11-s + (0.987 + 0.156i)12-s + (0.453 − 0.891i)13-s + (0.707 − 0.707i)14-s + (−0.156 + 0.987i)15-s + (0.309 − 0.951i)16-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 − 0.587i)4-s + (0.587 + 0.809i)5-s + (0.891 + 0.453i)6-s + (0.891 − 0.453i)7-s + (0.587 − 0.809i)8-s + i·9-s + (0.809 + 0.587i)10-s + (0.156 + 0.987i)11-s + (0.987 + 0.156i)12-s + (0.453 − 0.891i)13-s + (0.707 − 0.707i)14-s + (−0.156 + 0.987i)15-s + (0.309 − 0.951i)16-s + ⋯ |
Λ(s)=(=(697s/2ΓR(s+1)L(s)(0.771+0.636i)Λ(1−s)
Λ(s)=(=(697s/2ΓR(s+1)L(s)(0.771+0.636i)Λ(1−s)
Degree: |
1 |
Conductor: |
697
= 17⋅41
|
Sign: |
0.771+0.636i
|
Analytic conductor: |
74.9030 |
Root analytic conductor: |
74.9030 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ697(645,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 697, (1: ), 0.771+0.636i)
|
Particular Values
L(21) |
≈ |
6.686347304+2.404075790i |
L(21) |
≈ |
6.686347304+2.404075790i |
L(1) |
≈ |
3.006039521+0.5077531563i |
L(1) |
≈ |
3.006039521+0.5077531563i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 17 | 1 |
| 41 | 1 |
good | 2 | 1+(0.951−0.309i)T |
| 3 | 1+(0.707+0.707i)T |
| 5 | 1+(0.587+0.809i)T |
| 7 | 1+(0.891−0.453i)T |
| 11 | 1+(0.156+0.987i)T |
| 13 | 1+(0.453−0.891i)T |
| 19 | 1+(0.453+0.891i)T |
| 23 | 1+(0.309+0.951i)T |
| 29 | 1+(−0.987−0.156i)T |
| 31 | 1+(−0.809−0.587i)T |
| 37 | 1+(0.809−0.587i)T |
| 43 | 1+(0.951−0.309i)T |
| 47 | 1+(−0.891−0.453i)T |
| 53 | 1+(−0.987−0.156i)T |
| 59 | 1+(0.309+0.951i)T |
| 61 | 1+(0.951+0.309i)T |
| 67 | 1+(0.156−0.987i)T |
| 71 | 1+(−0.156−0.987i)T |
| 73 | 1+iT |
| 79 | 1+(−0.707−0.707i)T |
| 83 | 1+T |
| 89 | 1+(−0.891+0.453i)T |
| 97 | 1+(0.156−0.987i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.167055489730766629004087533861, −21.51214047710300195052111631032, −20.78602926882424472286218487508, −20.29545580862364005849615051212, −19.19727344175447675582958316127, −18.26400795160300167561181734707, −17.39251429797064524056297853926, −16.496765892509222168126895633160, −15.70315238701187186018138980459, −14.43134232430401220471105044519, −14.2455981826644227201767266771, −13.2406490645749164422013127159, −12.74930476786240773106009792462, −11.67442051309675680104647709446, −11.10530112359061373140087115395, −9.27570543142071467346968727355, −8.64733999097996606090059217371, −7.92707516317162983581031314796, −6.78388926478281431241181353637, −5.98079703071077455260078114882, −5.07595042449029192152006598648, −4.108925163409915208780753588467, −2.91247964868369763596439367014, −1.97176481252767565766453383779, −1.111058532942720267858376644640,
1.52615102941016876170297939175, 2.2554406239127848166145885533, 3.39575512857560767575750862297, 4.02744402196105064843839915014, 5.15657338503768115499627521617, 5.83668267983628717134340711024, 7.30275725181457729232270183489, 7.76944125621898257732676993186, 9.40090666814922889302949161853, 10.08968837199931321521730880159, 10.83850222846497275309184398311, 11.47363696732063334312624910675, 12.883217787947514481374925544629, 13.57064771260506847414327225740, 14.447841611494729787114669779254, 14.82948002639147495951623500995, 15.51333352656431645641278374987, 16.64070824937003715104970126076, 17.689516651103219235121729979472, 18.58377704398415343463685555068, 19.66812740714238808800039046277, 20.47922320329123952318448266203, 20.86547074517203371875761802501, 21.66273981188285652337046449502, 22.54947808816332264801000600832