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Properties

Label 2-2268-63.25-c1-0-17
Degree $2$
Conductor $2268$
Sign $0.966 - 0.257i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.433 − 0.750i)5-s + (2.32 − 1.26i)7-s + (1.75 + 3.04i)11-s + (0.933 + 1.61i)13-s + (3.25 − 5.64i)17-s + (2.69 + 4.66i)19-s + (−4.32 + 7.48i)23-s + (2.12 + 3.67i)25-s + (−1.75 + 3.04i)29-s − 1.86·31-s + (0.0576 − 2.29i)35-s + (1.39 + 2.40i)37-s + (−5.19 − 8.99i)41-s + (2.89 − 5.00i)43-s + 6.16·47-s + ⋯
L(s)  = 1  + (0.193 − 0.335i)5-s + (0.878 − 0.478i)7-s + (0.529 + 0.917i)11-s + (0.258 + 0.448i)13-s + (0.790 − 1.36i)17-s + (0.617 + 1.06i)19-s + (−0.901 + 1.56i)23-s + (0.424 + 0.735i)25-s + (−0.326 + 0.565i)29-s − 0.335·31-s + (0.00975 − 0.387i)35-s + (0.228 + 0.395i)37-s + (−0.810 − 1.40i)41-s + (0.440 − 0.763i)43-s + 0.898·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.966 - 0.257i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.966 - 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.221268796\)
\(L(\frac12)\) \(\approx\) \(2.221268796\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.32 + 1.26i)T \)
good5 \( 1 + (-0.433 + 0.750i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.75 - 3.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.933 - 1.61i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.25 + 5.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.69 - 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.32 - 7.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.75 - 3.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.86T + 31T^{2} \)
37 \( 1 + (-1.39 - 2.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.19 + 8.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.89 + 5.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.16T + 47T^{2} \)
53 \( 1 + (2.80 - 4.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.64T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + 1.35T + 67T^{2} \)
71 \( 1 - 2.08T + 71T^{2} \)
73 \( 1 + (3.62 - 6.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + (3.43 - 5.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.28 - 5.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.64 + 2.85i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.252319541354198811409339615272, −8.208661871914611398297090651962, −7.35938879593374639905666131060, −7.08575727976752613936693332841, −5.61561621194536274364643144542, −5.22607629042674384090148859654, −4.17533447331124647999269684518, −3.45597806276395045500000423571, −1.91231108368432095931619442053, −1.21606995324140912226646053161, 0.914044082635881364894824069893, 2.15715386940423920063301910394, 3.12584392163897679248954855784, 4.12994541311421590259753821535, 5.05093489572220188866976922867, 6.04229050583594634108255234169, 6.36424456138633843075677036952, 7.64747233684060161532905329399, 8.292265190169356939491917230487, 8.781729242647865469709510851492

Graph of the $Z$-function along the critical line