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Properties

Label 2-48e2-16.3-c0-0-1
Degree $2$
Conductor $2304$
Sign $0.923 - 0.382i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·7-s + (1 + i)13-s + (1.41 − 1.41i)19-s + i·25-s + 1.41i·31-s + (1 − i)37-s + (1.41 + 1.41i)43-s + 1.00·49-s + (1 + i)61-s − 1.41i·79-s + (−1.41 − 1.41i)91-s − 1.41·103-s + (−1 − i)109-s + ⋯
L(s)  = 1  − 1.41·7-s + (1 + i)13-s + (1.41 − 1.41i)19-s + i·25-s + 1.41i·31-s + (1 − i)37-s + (1.41 + 1.41i)43-s + 1.00·49-s + (1 + i)61-s − 1.41i·79-s + (−1.41 − 1.41i)91-s − 1.41·103-s + (−1 − i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.067831107\)
\(L(\frac12)\) \(\approx\) \(1.067831107\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 - iT^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.302546116607303680623599765129, −8.744298791460437827239012091672, −7.49042557300722951513499137485, −6.91151471657581457109635809315, −6.21638738816091816333599772854, −5.40470370449832701168055172066, −4.32170541358133313132859569341, −3.41496077221349515071698821322, −2.71009242995659255472354962937, −1.16102559576856923295065914169, 0.912462320279037600797760119232, 2.54787908494626515773169280733, 3.43752670980720403219442374935, 4.03671998590143400381670446504, 5.49310923906724657835834138844, 5.97381392879506404008184932827, 6.69617853735498492591926297991, 7.73988122232847371144935927056, 8.251088591587981408531724618665, 9.317603115592124857411980268424

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