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Properties

Label 2-224-32.13-c1-0-2
Degree 22
Conductor 224224
Sign 0.235−0.971i0.235 - 0.971i
Analytic cond. 1.788641.78864
Root an. cond. 1.337401.33740
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  + (−1.41 − 0.0115i)2-s + (−0.129 + 0.313i)3-s + (1.99 + 0.0327i)4-s + (−1.65 + 0.685i)5-s + (0.187 − 0.441i)6-s + (0.707 − 0.707i)7-s + (−2.82 − 0.0694i)8-s + (2.03 + 2.03i)9-s + (2.34 − 0.950i)10-s + (1.46 + 3.54i)11-s + (−0.269 + 0.622i)12-s + (−3.18 − 1.31i)13-s + (−1.00 + 0.991i)14-s − 0.608i·15-s + (3.99 + 0.130i)16-s + 5.24i·17-s + ⋯
L(s)  = 1  + (−0.999 − 0.00818i)2-s + (−0.0749 + 0.181i)3-s + (0.999 + 0.0163i)4-s + (−0.740 + 0.306i)5-s + (0.0764 − 0.180i)6-s + (0.267 − 0.267i)7-s + (−0.999 − 0.0245i)8-s + (0.679 + 0.679i)9-s + (0.742 − 0.300i)10-s + (0.442 + 1.06i)11-s + (−0.0779 + 0.179i)12-s + (−0.882 − 0.365i)13-s + (−0.269 + 0.265i)14-s − 0.157i·15-s + (0.999 + 0.0327i)16-s + 1.27i·17-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((0.235−0.971i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s+1/2) L(s)=((0.235−0.971i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: 0.235−0.971i0.235 - 0.971i
Analytic conductor: 1.788641.78864
Root analytic conductor: 1.337401.33740
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ224(141,⋅)\chi_{224} (141, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 224, ( :1/2), 0.235−0.971i)(2,\ 224,\ (\ :1/2),\ 0.235 - 0.971i)

Particular Values

L(1)L(1) ≈\approx 0.529715+0.416880i0.529715 + 0.416880i
L(12)L(\frac12) ≈\approx 0.529715+0.416880i0.529715 + 0.416880i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.41+0.0115i)T 1 + (1.41 + 0.0115i)T
7 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good3 1+(0.129−0.313i)T+(−2.12−2.12i)T2 1 + (0.129 - 0.313i)T + (-2.12 - 2.12i)T^{2}
5 1+(1.65−0.685i)T+(3.53−3.53i)T2 1 + (1.65 - 0.685i)T + (3.53 - 3.53i)T^{2}
11 1+(−1.46−3.54i)T+(−7.77+7.77i)T2 1 + (-1.46 - 3.54i)T + (-7.77 + 7.77i)T^{2}
13 1+(3.18+1.31i)T+(9.19+9.19i)T2 1 + (3.18 + 1.31i)T + (9.19 + 9.19i)T^{2}
17 1−5.24iT−17T2 1 - 5.24iT - 17T^{2}
19 1+(−0.490−0.203i)T+(13.4+13.4i)T2 1 + (-0.490 - 0.203i)T + (13.4 + 13.4i)T^{2}
23 1+(−5.03−5.03i)T+23iT2 1 + (-5.03 - 5.03i)T + 23iT^{2}
29 1+(1.56−3.76i)T+(−20.5−20.5i)T2 1 + (1.56 - 3.76i)T + (-20.5 - 20.5i)T^{2}
31 1−7.16T+31T2 1 - 7.16T + 31T^{2}
37 1+(−0.813+0.336i)T+(26.1−26.1i)T2 1 + (-0.813 + 0.336i)T + (26.1 - 26.1i)T^{2}
41 1+(5.83+5.83i)T+41iT2 1 + (5.83 + 5.83i)T + 41iT^{2}
43 1+(−0.0662−0.159i)T+(−30.4+30.4i)T2 1 + (-0.0662 - 0.159i)T + (-30.4 + 30.4i)T^{2}
47 1+11.3iT−47T2 1 + 11.3iT - 47T^{2}
53 1+(4.75+11.4i)T+(−37.4+37.4i)T2 1 + (4.75 + 11.4i)T + (-37.4 + 37.4i)T^{2}
59 1+(−11.2+4.64i)T+(41.7−41.7i)T2 1 + (-11.2 + 4.64i)T + (41.7 - 41.7i)T^{2}
61 1+(2.04−4.94i)T+(−43.1−43.1i)T2 1 + (2.04 - 4.94i)T + (-43.1 - 43.1i)T^{2}
67 1+(0.461−1.11i)T+(−47.3−47.3i)T2 1 + (0.461 - 1.11i)T + (-47.3 - 47.3i)T^{2}
71 1+(0.00269−0.00269i)T−71iT2 1 + (0.00269 - 0.00269i)T - 71iT^{2}
73 1+(−2.92−2.92i)T+73iT2 1 + (-2.92 - 2.92i)T + 73iT^{2}
79 1+7.96iT−79T2 1 + 7.96iT - 79T^{2}
83 1+(6.66+2.76i)T+(58.6+58.6i)T2 1 + (6.66 + 2.76i)T + (58.6 + 58.6i)T^{2}
89 1+(2.79−2.79i)T−89iT2 1 + (2.79 - 2.79i)T - 89iT^{2}
97 1−16.0T+97T2 1 - 16.0T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−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

−12.15643201652121429419794873279, −11.32023770788148778989006290349, −10.34643440949824235101272678760, −9.772428713839357297429533317475, −8.414489583701788032760650080409, −7.45283494231335476668792491328, −6.93540486921475984146230616438, −5.12537203157789210999841639350, −3.67921423030455831836300220592, −1.82691976295309931830506841079, 0.819643404740775601678555730480, 2.87445120760820852139278729719, 4.56038538318944091262394926955, 6.24415231796349565778800637432, 7.18505355319716828505667229374, 8.152902349465165965862794399632, 9.091134292274302939922175224208, 9.887831910967790833970763552489, 11.26687104917174148437405462932, 11.80587442157087844738314514480

Graph of the ZZ-function along the critical line