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Properties

Label 2-2268-63.5-c1-0-12
Degree $2$
Conductor $2268$
Sign $0.236 - 0.971i$
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  + (1.00 + 1.73i)5-s + (1.26 + 2.32i)7-s + (4.15 + 2.40i)11-s + (0.864 + 0.499i)13-s + (−0.0445 − 0.0772i)17-s + (3.68 + 2.12i)19-s + (−0.839 + 0.484i)23-s + (0.492 − 0.853i)25-s + (2.93 − 1.69i)29-s − 5.03i·31-s + (−2.76 + 4.52i)35-s + (−0.0675 + 0.117i)37-s + (−5.62 + 9.73i)41-s + (−3.66 − 6.35i)43-s + 3.53·47-s + ⋯
L(s)  = 1  + (0.447 + 0.775i)5-s + (0.478 + 0.878i)7-s + (1.25 + 0.723i)11-s + (0.239 + 0.138i)13-s + (−0.0108 − 0.0187i)17-s + (0.846 + 0.488i)19-s + (−0.175 + 0.101i)23-s + (0.0985 − 0.170i)25-s + (0.544 − 0.314i)29-s − 0.903i·31-s + (−0.467 + 0.764i)35-s + (−0.0111 + 0.0192i)37-s + (−0.877 + 1.52i)41-s + (−0.559 − 0.968i)43-s + 0.515·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.236 - 0.971i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.236 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.300101258\)
\(L(\frac12)\) \(\approx\) \(2.300101258\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.26 - 2.32i)T \)
good5 \( 1 + (-1.00 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.15 - 2.40i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.864 - 0.499i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.0445 + 0.0772i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.68 - 2.12i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.839 - 0.484i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.93 + 1.69i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.03iT - 31T^{2} \)
37 \( 1 + (0.0675 - 0.117i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.62 - 9.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.66 + 6.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.53T + 47T^{2} \)
53 \( 1 + (5.31 - 3.07i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.76T + 59T^{2} \)
61 \( 1 + 13.0iT - 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 - 4.56iT - 71T^{2} \)
73 \( 1 + (-3.73 + 2.15i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + (1.62 + 2.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.06 + 1.84i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.03 + 0.597i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.331492583251676430653546449076, −8.427760560817199021784669998712, −7.65989752006422113255463259223, −6.66249538935463821617717823604, −6.22937269309012836875299518024, −5.29295156663023620017909696647, −4.36728247459384600374669589808, −3.35108788534024070693803531642, −2.32825770317584215786370404465, −1.46329895547990674699812204493, 0.900169750999864159440466176742, 1.56702812596211856776189886576, 3.16570690454290476745240568119, 4.00871989977198978661983814979, 4.89806362066259752394973059593, 5.59853929553817032729739649266, 6.60997773851751254527777681455, 7.22517112819633831582826701713, 8.284154517741891900983700462627, 8.832961470900905266864535643015

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