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Properties

Label 2-3332-476.67-c0-0-7
Degree 22
Conductor 33323332
Sign −0.266−0.963i-0.266 - 0.963i
Analytic cond. 1.662881.66288
Root an. cond. 1.289521.28952
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.999·8-s + (0.5 + 0.866i)9-s + 2·13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + (−0.5 + 0.866i)25-s + (1 + 1.73i)26-s + (0.499 − 0.866i)32-s + 0.999·34-s − 0.999·36-s − 0.999·50-s + (−0.999 + 1.73i)52-s + (−1 + 1.73i)53-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.999·8-s + (0.5 + 0.866i)9-s + 2·13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + (−0.5 + 0.866i)25-s + (1 + 1.73i)26-s + (0.499 − 0.866i)32-s + 0.999·34-s − 0.999·36-s − 0.999·50-s + (−0.999 + 1.73i)52-s + (−1 + 1.73i)53-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((−0.266−0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((−0.266−0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: −0.266−0.963i-0.266 - 0.963i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(67,⋅)\chi_{3332} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), −0.266−0.963i)(2,\ 3332,\ (\ :0),\ -0.266 - 0.963i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6987124861.698712486
L(12)L(\frac12) ≈\approx 1.6987124861.698712486
L(1)L(1) 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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1 1
17 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good3 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
5 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
13 1−2T+T2 1 - 2T + T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+T2 1 + T^{2}
73 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
79 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
97 1−T2 1 - T^{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

−8.870431060253199522193022083744, −8.052473101106199130887091473846, −7.56244060526558704183467388904, −6.76629422866384744025966600858, −5.95236142671556961796155853963, −5.36332984954896989788059762674, −4.47887448849527279390066982580, −3.72845319858894840005191816088, −2.87969008200034055028423249611, −1.43444400753966688670149187750, 1.02368798150210056960391418729, 1.87312827483466508110981069627, 3.24897868620786090643291514224, 3.77334980129548481202183683420, 4.43172519368160728415277307891, 5.62264660835330541231651592472, 6.17516738020270335602904850429, 6.78421467486257145275805207514, 8.176187904380130979774471702948, 8.612420346822928727030383580695

Graph of the ZZ-function along the critical line