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Properties

Label 2-288-96.35-c1-0-13
Degree $2$
Conductor $288$
Sign $-0.911 + 0.410i$
Analytic cond. $2.29969$
Root an. cond. $1.51647$
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  + (−1.19 + 0.752i)2-s + (0.867 − 1.80i)4-s + (−0.0913 + 0.0378i)5-s + (−3.05 − 3.05i)7-s + (0.317 + 2.81i)8-s + (0.0808 − 0.114i)10-s + (−5.25 + 2.17i)11-s + (−1.57 + 3.79i)13-s + (5.95 + 1.35i)14-s + (−2.49 − 3.12i)16-s − 2.56·17-s + (2.64 + 1.09i)19-s + (−0.0110 + 0.197i)20-s + (4.65 − 6.56i)22-s + (−4.03 − 4.03i)23-s + ⋯
L(s)  = 1  + (−0.846 + 0.532i)2-s + (0.433 − 0.901i)4-s + (−0.0408 + 0.0169i)5-s + (−1.15 − 1.15i)7-s + (0.112 + 0.993i)8-s + (0.0255 − 0.0360i)10-s + (−1.58 + 0.656i)11-s + (−0.436 + 1.05i)13-s + (1.59 + 0.363i)14-s + (−0.623 − 0.781i)16-s − 0.622·17-s + (0.605 + 0.250i)19-s + (−0.00246 + 0.0441i)20-s + (0.992 − 1.39i)22-s + (−0.840 − 0.840i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.911 + 0.410i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.911 + 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00812444 - 0.0378294i\)
\(L(\frac12)\) \(\approx\) \(0.00812444 - 0.0378294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 - 0.752i)T \)
3 \( 1 \)
good5 \( 1 + (0.0913 - 0.0378i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (3.05 + 3.05i)T + 7iT^{2} \)
11 \( 1 + (5.25 - 2.17i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.57 - 3.79i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 2.56T + 17T^{2} \)
19 \( 1 + (-2.64 - 1.09i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.03 + 4.03i)T + 23iT^{2} \)
29 \( 1 + (-2.06 + 4.99i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.44iT - 31T^{2} \)
37 \( 1 + (2.07 + 5.01i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.296 + 0.296i)T - 41iT^{2} \)
43 \( 1 + (2.72 + 6.57i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 7.42iT - 47T^{2} \)
53 \( 1 + (-1.53 - 3.69i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.988 - 2.38i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-10.4 - 4.32i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (0.690 - 1.66i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (2.97 - 2.97i)T - 71iT^{2} \)
73 \( 1 + (-9.22 - 9.22i)T + 73iT^{2} \)
79 \( 1 + 1.12T + 79T^{2} \)
83 \( 1 + (4.13 - 9.98i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (12.0 + 12.0i)T + 89iT^{2} \)
97 \( 1 + 18.5T + 97T^{2} \)
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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

−11.07117576117558623909398297588, −10.00199948222427861313143117315, −9.863608050598072747952547196473, −8.462139106465272791201010082978, −7.31927968202249040121120526992, −6.88985072587752359521950559428, −5.57567075627652956033034805218, −4.18916854672746344340409521291, −2.33100020829833466290797148822, −0.03430375563084707151448133590, 2.53329256693875890704895559915, 3.25236760238888188731059452523, 5.30872868183554521193441668421, 6.37508968507399622765189299622, 7.75984267685318208742595970226, 8.438612793754524257842450753920, 9.570436989028757522189652761606, 10.15287579371062853447195506872, 11.18090749335353831674315034089, 12.18322556733156791362240592674

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