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Properties

Label 2-2268-63.4-c1-0-26
Degree $2$
Conductor $2268$
Sign $0.296 + 0.954i$
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  + (2.5 + 0.866i)7-s + (3.5 − 6.06i)13-s + (−4 − 6.92i)19-s − 5·25-s + (−5.5 − 9.52i)31-s + (0.5 + 0.866i)37-s + (6.5 + 11.2i)43-s + (5.5 + 4.33i)49-s + (0.5 − 0.866i)61-s + (−5.5 − 9.52i)67-s + (5 − 8.66i)73-s + (6.5 − 11.2i)79-s + (14 − 12.1i)91-s + (9.5 + 16.4i)97-s − 7·103-s + ⋯
L(s)  = 1  + (0.944 + 0.327i)7-s + (0.970 − 1.68i)13-s + (−0.917 − 1.58i)19-s − 25-s + (−0.987 − 1.71i)31-s + (0.0821 + 0.142i)37-s + (0.991 + 1.71i)43-s + (0.785 + 0.618i)49-s + (0.0640 − 0.110i)61-s + (−0.671 − 1.16i)67-s + (0.585 − 1.01i)73-s + (0.731 − 1.26i)79-s + (1.46 − 1.27i)91-s + (0.964 + 1.67i)97-s − 0.689·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.296 + 0.954i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.296 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.755963055\)
\(L(\frac12)\) \(\approx\) \(1.755963055\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-3.5 + 6.06i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.5 + 9.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.5 - 11.2i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
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

−8.855654184481030745096709376121, −7.911638068117581806348353716587, −7.70251209612438451623131648381, −6.34214905477852995336998179975, −5.75500431826816915165607452392, −4.90005578062616256892419122176, −4.05127395161586616054015201291, −2.94733933372954155163963973555, −1.98144627595263414713135275594, −0.61604625888495184504997621295, 1.44263098814655918152925038770, 2.07593785656695397395522205076, 3.82790381884431799385968421150, 4.07604862199340948940615547105, 5.24144203248153702615291336161, 6.05583712084885767481934208477, 6.89493678956049389491769496479, 7.64143415997280399048965801694, 8.570818511531859968699547558832, 8.902678648353312107301548436596

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