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Properties

Label 2-2816-8.5-c1-0-58
Degree $2$
Conductor $2816$
Sign $0.707 + 0.707i$
Analytic cond. $22.4858$
Root an. cond. $4.74192$
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  − i·3-s + 3i·5-s + 2·7-s + 2·9-s − i·11-s − 4i·13-s + 3·15-s + 6·17-s − 8i·19-s − 2i·21-s − 3·23-s − 4·25-s − 5i·27-s − 5·31-s − 33-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.34i·5-s + 0.755·7-s + 0.666·9-s − 0.301i·11-s − 1.10i·13-s + 0.774·15-s + 1.45·17-s − 1.83i·19-s − 0.436i·21-s − 0.625·23-s − 0.800·25-s − 0.962i·27-s − 0.898·31-s − 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2816\)    =    \(2^{8} \cdot 11\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(22.4858\)
Root analytic conductor: \(4.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2816} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2816,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.159553409\)
\(L(\frac12)\) \(\approx\) \(2.159553409\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + iT \)
good3 \( 1 + iT - 3T^{2} \)
5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 3iT - 59T^{2} \)
61 \( 1 + 4iT - 61T^{2} \)
67 \( 1 - iT - 67T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−8.479735901623889512906431104637, −7.68058570534455213905345769922, −7.31437092181870441550197694087, −6.58642296438673709868121730040, −5.70678604533203336778169784331, −4.91917011087619770470001176879, −3.71237377146158915222819033021, −2.93246042476680398511339972023, −2.02756228663983482607334024776, −0.76972859212508805144963254017, 1.29864834843973537807673569952, 1.81732056715599726184422188859, 3.65813729379500155857067918963, 4.18071973290715345880449931197, 4.98580635441500194695531168488, 5.49881454450631981897258445007, 6.54952554932924157465150224486, 7.77613016440476278051664858110, 8.008311562577221525516860548542, 8.986275376550765090473696063540

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