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Properties

Label 2-468-117.103-c1-0-6
Degree $2$
Conductor $468$
Sign $0.379 - 0.925i$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
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  + (0.444 + 1.67i)3-s + (2.13 + 1.23i)5-s + (0.836 − 0.483i)7-s + (−2.60 + 1.48i)9-s + (3.21 − 1.85i)11-s + (3.60 + 0.128i)13-s + (−1.11 + 4.12i)15-s − 2.85·17-s + 0.604i·19-s + (1.18 + 1.18i)21-s + (−3.15 + 5.46i)23-s + (0.548 + 0.949i)25-s + (−3.64 − 3.69i)27-s + (0.550 + 0.954i)29-s + (−8.22 − 4.75i)31-s + ⋯
L(s)  = 1  + (0.256 + 0.966i)3-s + (0.956 + 0.552i)5-s + (0.316 − 0.182i)7-s + (−0.868 + 0.495i)9-s + (0.969 − 0.559i)11-s + (0.999 + 0.0355i)13-s + (−0.288 + 1.06i)15-s − 0.691·17-s + 0.138i·19-s + (0.257 + 0.258i)21-s + (−0.658 + 1.14i)23-s + (0.109 + 0.189i)25-s + (−0.702 − 0.711i)27-s + (0.102 + 0.177i)29-s + (−1.47 − 0.853i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.379 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53977 + 1.03267i\)
\(L(\frac12)\) \(\approx\) \(1.53977 + 1.03267i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.444 - 1.67i)T \)
13 \( 1 + (-3.60 - 0.128i)T \)
good5 \( 1 + (-2.13 - 1.23i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.836 + 0.483i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.21 + 1.85i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.85T + 17T^{2} \)
19 \( 1 - 0.604iT - 19T^{2} \)
23 \( 1 + (3.15 - 5.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.550 - 0.954i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.22 + 4.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.98iT - 37T^{2} \)
41 \( 1 + (-10.2 - 5.92i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.32 + 7.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.83 - 1.63i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.79T + 53T^{2} \)
59 \( 1 + (-5.40 - 3.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.03 - 6.98i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.70 + 4.45i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 + 13.9iT - 73T^{2} \)
79 \( 1 + (4.13 + 7.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.62 - 0.939i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.377iT - 89T^{2} \)
97 \( 1 + (-5.89 + 3.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.05721693652523138243743944109, −10.32926976969246804927603830921, −9.355827101792753016182772262254, −8.853504102260593346763544671159, −7.64793073024422102859222134199, −6.24024297792270456668243332975, −5.69850053535200559297078014506, −4.25239159424046289637969608026, −3.37895884874893101934391964139, −1.90826050279660351876337643244, 1.34122663588536324296364887729, 2.27530107657032866699269691274, 3.96670689581773596221451868501, 5.37293932346239655014990151740, 6.29476632992854820865465426827, 7.02125822579774881005439517275, 8.375335901898964235126553694376, 8.883166345630694479452579965990, 9.752682153299211888142306396861, 11.03005084049972066488634995497

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