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Properties

Label 2-2268-63.58-c1-0-24
Degree $2$
Conductor $2268$
Sign $-0.580 + 0.814i$
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  + (−0.5 − 2.59i)7-s + (−1 + 1.73i)13-s + (0.5 − 0.866i)19-s + (2.5 − 4.33i)25-s − 7·31-s + (5 − 8.66i)37-s + (−2.5 − 4.33i)43-s + (−6.5 + 2.59i)49-s − 61-s − 16·67-s + (−8.5 − 14.7i)73-s − 4·79-s + (5 + 1.73i)91-s + (9.5 + 16.4i)97-s + (−10 − 17.3i)103-s + ⋯
L(s)  = 1  + (−0.188 − 0.981i)7-s + (−0.277 + 0.480i)13-s + (0.114 − 0.198i)19-s + (0.5 − 0.866i)25-s − 1.25·31-s + (0.821 − 1.42i)37-s + (−0.381 − 0.660i)43-s + (−0.928 + 0.371i)49-s − 0.128·61-s − 1.95·67-s + (−0.994 − 1.72i)73-s − 0.450·79-s + (0.524 + 0.181i)91-s + (0.964 + 1.67i)97-s + (−0.985 − 1.70i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.580 + 0.814i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.007679373\)
\(L(\frac12)\) \(\approx\) \(1.007679373\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 16T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (8.5 + 14.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.5 - 16.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.888572742096398216758273610861, −7.81712759934282365097757900860, −7.24622133768332024098581760994, −6.53477433935067566432476937950, −5.62253722790853836220016482228, −4.58874496882461498747731741089, −3.95157708761761045975569086885, −2.92378891417182228722388778675, −1.72349709382736545315948360924, −0.34344478169413923337686489562, 1.44810529292625391476818320843, 2.65683191394103980415419442736, 3.37661580573769109570361439991, 4.60013963629685721316710159191, 5.42116715680561625516084718843, 6.05482339964846725006141148652, 6.98526001907009710630894561713, 7.79981764771986420062339712429, 8.577333052580492898044250277417, 9.266767651967518653346250148897

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