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Properties

Label 2-2268-63.5-c1-0-25
Degree $2$
Conductor $2268$
Sign $-0.141 + 0.989i$
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 + (−1.5 − 0.866i)13-s + (3 + 1.73i)19-s + (2.5 − 4.33i)25-s − 1.73i·31-s + (−0.5 + 0.866i)37-s + (−6.5 − 11.2i)43-s + (−6.5 − 2.59i)49-s − 15.5i·61-s + 11·67-s + (−12 + 6.92i)73-s − 13·79-s + (−3 + 3.46i)91-s + (4.5 − 2.59i)97-s + (16.5 − 9.52i)103-s + ⋯
L(s)  = 1  + (0.188 − 0.981i)7-s + (−0.416 − 0.240i)13-s + (0.688 + 0.397i)19-s + (0.5 − 0.866i)25-s − 0.311i·31-s + (−0.0821 + 0.142i)37-s + (−0.991 − 1.71i)43-s + (−0.928 − 0.371i)49-s − 1.99i·61-s + 1.34·67-s + (−1.40 + 0.810i)73-s − 1.46·79-s + (−0.314 + 0.363i)91-s + (0.456 − 0.263i)97-s + (1.62 − 0.938i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.141 + 0.989i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.141 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.420761493\)
\(L(\frac12)\) \(\approx\) \(1.420761493\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
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 \( 1 + 47T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15.5iT - 61T^{2} \)
67 \( 1 - 11T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (12 - 6.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.5 + 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.694748899908788055745499889006, −8.019249613283833509541588126839, −7.25182491726101085399992354996, −6.64528674844189043892446630343, −5.58342178553037669370649361847, −4.78494597310027743368799352860, −3.93027957803695273354036887392, −3.04216336481633787324025528550, −1.78380373860135650601105111534, −0.50382477383887372135006834477, 1.36909543619551245611186742485, 2.53066720823162018393243113528, 3.32459527405826226300462766013, 4.59696857323131445477521214691, 5.24628324518402724872483446401, 6.04630490301077515619014444170, 6.94927096091884629168902746218, 7.68952909875811283598634950756, 8.578712919384978819319673265304, 9.178897769789374840228326136303

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