Properties

Label 2-1216-152.75-c1-0-25
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  − 1.12i·3-s + 0.0952i·7-s + 1.72·9-s + 6.35·13-s − 4.53·17-s − 4.35i·19-s + 0.107·21-s + 9.35i·23-s + 5·25-s − 5.33i·27-s − 4.09·29-s + 8.71·37-s − 7.16i·39-s − 6i·47-s + 6.99·49-s + ⋯
L(s)  = 1  − 0.651i·3-s + 0.0360i·7-s + 0.575·9-s + 1.76·13-s − 1.10·17-s − 0.999i·19-s + 0.0234·21-s + 1.95i·23-s + 25-s − 1.02i·27-s − 0.760·29-s + 1.43·37-s − 1.14i·39-s − 0.875i·47-s + 0.998·49-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: $\chi_{1216} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.847190321\)
\(L(\frac12)\) \(\approx\) \(1.847190321\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.35iT \)
good3 \( 1 + 1.12iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 0.0952iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.35T + 13T^{2} \)
17 \( 1 + 4.53T + 17T^{2} \)
23 \( 1 - 9.35iT - 23T^{2} \)
29 \( 1 + 4.09T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 8.71T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + 11.5iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 16.0iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.420590300011157922780530716071, −8.862480050657730083160453033771, −7.925327301794570934130660008428, −7.08925819519704063987292632205, −6.45194329261096817876321663820, −5.53888091236379014148721565917, −4.37228381782609191746803685056, −3.45682292045642817384465841761, −2.09017692081194110916467871165, −1.00442505296506674979221834573, 1.21310877272745023636475915826, 2.68627560934883496493254520034, 4.04809674069850140358972213701, 4.32610679317451350393578730471, 5.67664823808770301655670714003, 6.43618783404534822832572486524, 7.31767527618999705664466064113, 8.558832372606625280775071292550, 8.841981572187319847586600192375, 9.944860121109089065726380780328

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