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Properties

Label 2-48e2-3.2-c0-0-0
Degree $2$
Conductor $2304$
Sign $-0.577 - 0.816i$
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  + 1.41i·5-s − 2·13-s + 1.41i·17-s − 1.00·25-s + 1.41i·29-s − 1.41i·41-s − 49-s + 1.41i·53-s − 2.82i·65-s − 2.00·85-s + 1.41i·89-s − 1.41i·101-s + 2·109-s + 1.41i·113-s + ⋯
L(s)  = 1  + 1.41i·5-s − 2·13-s + 1.41i·17-s − 1.00·25-s + 1.41i·29-s − 1.41i·41-s − 49-s + 1.41i·53-s − 2.82i·65-s − 2.00·85-s + 1.41i·89-s − 1.41i·101-s + 2·109-s + 1.41i·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8319996556\)
\(L(\frac12)\) \(\approx\) \(0.8319996556\)
\(L(1)\) 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 \( 1 - 1.41iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 2T + T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
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

−9.600382519314067409425240834359, −8.694713240169931339343211450585, −7.67525022228218376117662889250, −7.17342791169523369123507386953, −6.51025095753160281438593897872, −5.61354566421476800335894918704, −4.67366925330211298521715875519, −3.61422743557924062040303804826, −2.79107145983021744794964407166, −1.92915414754702691980789632561, 0.53493052033900144993726839620, 2.00437913210693491302489167006, 2.99430039106863727725013774060, 4.45382301204594312228857793787, 4.83630017757732207590779184401, 5.52686335555049534084164717510, 6.66806023330649798218069321958, 7.55071859520826889512189049595, 8.106172408262329228236930236965, 9.030542698482044622429187036784

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