Properties

Label 2-1216-8.5-c1-0-18
Degree $2$
Conductor $1216$
Sign $-0.707 - 0.707i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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  + 2i·3-s + 3.31i·5-s + 3.31·7-s − 9-s + 5i·11-s − 6.63·15-s + 5·17-s i·19-s + 6.63i·21-s + 6.63·23-s − 6·25-s + 4i·27-s − 6.63i·29-s − 10·33-s + 11i·35-s + ⋯
L(s)  = 1  + 1.15i·3-s + 1.48i·5-s + 1.25·7-s − 0.333·9-s + 1.50i·11-s − 1.71·15-s + 1.21·17-s − 0.229i·19-s + 1.44i·21-s + 1.38·23-s − 1.20·25-s + 0.769i·27-s − 1.23i·29-s − 1.74·33-s + 1.85i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.055327798\)
\(L(\frac12)\) \(\approx\) \(2.055327798\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + iT \)
good3 \( 1 - 2iT - 3T^{2} \)
5 \( 1 - 3.31iT - 5T^{2} \)
7 \( 1 - 3.31T + 7T^{2} \)
11 \( 1 - 5iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
23 \( 1 - 6.63T + 23T^{2} \)
29 \( 1 + 6.63iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6.63iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 9.94T + 47T^{2} \)
53 \( 1 + 13.2iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 9.94iT - 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 6.63T + 71T^{2} \)
73 \( 1 + 9T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−10.01744647451125864909654590107, −9.615221575350140801979869938460, −8.425148354998472613330561997437, −7.36263590671309061372011551181, −7.03597694554568477782925793613, −5.61312760757611456318205966009, −4.80729650171502837973667391370, −4.03421058638350864897326894513, −3.01287100473840157779367045948, −1.86826817566729217909127306176, 1.17055033005542009634975040100, 1.26258727654046097602948520640, 3.00870965464448832371541584128, 4.42154850135880767583686358960, 5.30005584770578324381022678625, 5.89138904798455870273502042289, 7.18387726210166083615808742675, 7.87429309009036761473761238422, 8.591429640219863182807935845033, 8.929856899491570786025198779023

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