Properties

Label 2-1216-8.5-c1-0-4
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  + 3i·3-s + 4i·5-s + 7-s − 6·9-s − 5i·13-s − 12·15-s − 5·17-s + i·19-s + 3i·21-s − 3·23-s − 11·25-s − 9i·27-s + 7i·29-s + 10·31-s + 4i·35-s + ⋯
L(s)  = 1  + 1.73i·3-s + 1.78i·5-s + 0.377·7-s − 2·9-s − 1.38i·13-s − 3.09·15-s − 1.21·17-s + 0.229i·19-s + 0.654i·21-s − 0.625·23-s − 2.20·25-s − 1.73i·27-s + 1.29i·29-s + 1.79·31-s + 0.676i·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

\(L(1)\) \(\approx\) \(1.101665486\)
\(L(\frac12)\) \(\approx\) \(1.101665486\)
\(L(\frac{3}{2})\) 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 \( 1 - 3iT - 3T^{2} \)
5 \( 1 - 4iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 - 7iT - 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 - iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 - 7iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
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

−10.34162321963084756228495507212, −9.789944851281245100404629534036, −8.702666780100205789920154436905, −7.931113381001288443833542932298, −6.82861611084092418158079627366, −5.97086886665497760988066145289, −5.06095169685955457986079072353, −4.07736104185211590370685136133, −3.23270879701863519368115653074, −2.55192185855878262092093367963, 0.45615319848583992520295124011, 1.59768397922571576498652599915, 2.25676566650142583540662821214, 4.23343298498192265577536023043, 4.92058352265273508167847449905, 6.09620023331127049825297063168, 6.66782394428390095259465866418, 7.74981057464855372059181133882, 8.349699421063129414052767235301, 8.888684763342553795967179645308

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