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Properties

Label 2-338-13.12-c1-0-5
Degree 22
Conductor 338338
Sign 0.0304−0.999i0.0304 - 0.999i
Analytic cond. 2.698942.69894
Root an. cond. 1.642841.64284
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  + i·2-s + 2.35·3-s − 4-s + 0.890i·5-s + 2.35i·6-s + 4.49i·7-s − i·8-s + 2.55·9-s − 0.890·10-s − 2.69i·11-s − 2.35·12-s − 4.49·14-s + 2.09i·15-s + 16-s − 3.58·17-s + 2.55i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.36·3-s − 0.5·4-s + 0.398i·5-s + 0.962i·6-s + 1.69i·7-s − 0.353i·8-s + 0.851·9-s − 0.281·10-s − 0.811i·11-s − 0.680·12-s − 1.20·14-s + 0.541i·15-s + 0.250·16-s − 0.868·17-s + 0.602i·18-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=((0.0304−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0304 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+1/2) L(s)=((0.0304−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0304 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: 0.0304−0.999i0.0304 - 0.999i
Analytic conductor: 2.698942.69894
Root analytic conductor: 1.642841.64284
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ338(337,⋅)\chi_{338} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 338, ( :1/2), 0.0304−0.999i)(2,\ 338,\ (\ :1/2),\ 0.0304 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.34219+1.30188i1.34219 + 1.30188i
L(12)L(\frac12) ≈\approx 1.34219+1.30188i1.34219 + 1.30188i
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−iT 1 - iT
13 1 1
good3 1−2.35T+3T2 1 - 2.35T + 3T^{2}
5 1−0.890iT−5T2 1 - 0.890iT - 5T^{2}
7 1−4.49iT−7T2 1 - 4.49iT - 7T^{2}
11 1+2.69iT−11T2 1 + 2.69iT - 11T^{2}
17 1+3.58T+17T2 1 + 3.58T + 17T^{2}
19 1+2.93iT−19T2 1 + 2.93iT - 19T^{2}
23 1−6.09T+23T2 1 - 6.09T + 23T^{2}
29 1−2.98T+29T2 1 - 2.98T + 29T^{2}
31 1+2.39iT−31T2 1 + 2.39iT - 31T^{2}
37 1−1.50iT−37T2 1 - 1.50iT - 37T^{2}
41 1+3.65iT−41T2 1 + 3.65iT - 41T^{2}
43 1+0.170T+43T2 1 + 0.170T + 43T^{2}
47 1+5.20iT−47T2 1 + 5.20iT - 47T^{2}
53 1−6.09T+53T2 1 - 6.09T + 53T^{2}
59 1+3.07iT−59T2 1 + 3.07iT - 59T^{2}
61 1+13.9T+61T2 1 + 13.9T + 61T^{2}
67 1−11.0iT−67T2 1 - 11.0iT - 67T^{2}
71 1+10.0iT−71T2 1 + 10.0iT - 71T^{2}
73 1+10.9iT−73T2 1 + 10.9iT - 73T^{2}
79 1−2.81T+79T2 1 - 2.81T + 79T^{2}
83 1−2.93iT−83T2 1 - 2.93iT - 83T^{2}
89 1+12.1iT−89T2 1 + 12.1iT - 89T^{2}
97 1−12.9iT−97T2 1 - 12.9iT - 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

−11.85640433446130502167651413798, −10.77504573192763220100176117260, −9.270824225578546871512975247830, −8.882270577871141719572348289069, −8.282874339280578244700172004824, −7.06726461701306438080338256548, −6.05068560950295705100068242991, −4.89404505612156742491595940295, −3.24439438021732674384514175994, −2.44457143953595459335701192797, 1.36532783804410111046656701360, 2.86455585472134964059290587836, 3.98611647356628423337514117727, 4.75134476885713122634305475743, 6.86236379213455161399702346428, 7.70569317492863907635810165615, 8.662247785862802145917985596001, 9.470068944905294044316091223976, 10.33242152266622065880616166804, 11.09321996409962345467663912464

Graph of the ZZ-function along the critical line