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Properties

Label 2-2268-63.47-c1-0-26
Degree $2$
Conductor $2268$
Sign $-0.471 + 0.881i$
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 + (4.5 − 2.59i)13-s + (−7.5 − 4.33i)19-s − 5·25-s + (1.5 + 0.866i)31-s + (5.5 − 9.52i)37-s + (−6.5 + 11.2i)43-s + (−6.5 − 2.59i)49-s + (6 − 3.46i)61-s + (−2.5 + 4.33i)67-s + (−1.5 + 0.866i)73-s + (−8.5 − 14.7i)79-s + (−4.5 − 12.9i)91-s + (−12 − 6.92i)97-s − 15.5i·103-s + ⋯
L(s)  = 1  + (0.188 − 0.981i)7-s + (1.24 − 0.720i)13-s + (−1.72 − 0.993i)19-s − 25-s + (0.269 + 0.155i)31-s + (0.904 − 1.56i)37-s + (−0.991 + 1.71i)43-s + (−0.928 − 0.371i)49-s + (0.768 − 0.443i)61-s + (−0.305 + 0.529i)67-s + (−0.175 + 0.101i)73-s + (−0.956 − 1.65i)79-s + (−0.471 − 1.36i)91-s + (−1.21 − 0.703i)97-s − 1.53i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.471 + 0.881i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.471 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.302108621\)
\(L(\frac12)\) \(\approx\) \(1.302108621\)
\(L(\frac{3}{2})\) 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 \( 1 + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.5 + 4.33i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \)
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 \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (1.5 - 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12 + 6.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.591136329343221754299480144843, −8.101643497405402838011373242662, −7.23374981002689541370255697442, −6.41894647877747878393988748706, −5.74894934170125048554235184319, −4.54282694794454783308357518015, −3.99835713751433763451568869440, −2.96732789724893050138109219792, −1.69743194326007219384634241561, −0.44082767595949334476307539968, 1.55344476624675286745246121697, 2.37998353963323020581031477000, 3.65301764510918033427104555916, 4.35629924131211705834691894046, 5.46854253744808558117497816264, 6.18706164906192565972816374562, 6.71330736461364241783891291740, 8.084623541462540238303941578809, 8.408645743331234944141388432961, 9.140531662285794971500046615681

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