Properties

Label 2-1216-152.37-c0-0-3
Degree $2$
Conductor $1216$
Sign $0.707 + 0.707i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
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  − 9-s − 2i·11-s + 2·17-s i·19-s + 25-s + 2i·43-s − 49-s − 2·73-s + 81-s − 2i·83-s + 2i·99-s + ⋯
L(s)  = 1  − 9-s − 2i·11-s + 2·17-s i·19-s + 25-s + 2i·43-s − 49-s − 2·73-s + 81-s − 2i·83-s + 2i·99-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.004877544\)
\(L(\frac12)\) \(\approx\) \(1.004877544\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.788327927997988828700733405486, −8.846577655230746483293077608280, −8.332150890540909515735825309940, −7.51386320559291969345131698271, −6.25276101971310853271943309746, −5.74416293752405322785256604644, −4.84572381246513044076514900335, −3.27243948113723452794469111396, −2.98582072277298909962376996114, −0.973135725108569989388929430460, 1.62813312193582553836094851059, 2.86225146813457197621463621780, 3.90882605367159703739915161423, 5.06060394405552271932057231691, 5.69307302927373193768143545120, 6.83392596893603686582455254946, 7.60113094711622986676220564521, 8.318244500592858537973108643467, 9.336471726877595065706323718001, 10.03858683488733557500576328186

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