Properties

Label 2-1216-152.75-c1-0-39
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  − 3.40i·3-s − 4.62i·7-s − 8.56·9-s + 0.223·13-s − 3.69·17-s + 4.35i·19-s − 15.7·21-s − 6.49i·23-s + 5·25-s + 18.9i·27-s + 6.57·29-s − 8.71·37-s − 0.761i·39-s − 6i·47-s − 14.4·49-s + ⋯
L(s)  = 1  − 1.96i·3-s − 1.74i·7-s − 2.85·9-s + 0.0620·13-s − 0.896·17-s + 0.999i·19-s − 3.43·21-s − 1.35i·23-s + 25-s + 3.64i·27-s + 1.22·29-s − 1.43·37-s − 0.121i·39-s − 0.875i·47-s − 2.06·49-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

\(L(1)\) \(\approx\) \(1.072439141\)
\(L(\frac12)\) \(\approx\) \(1.072439141\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.40iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 4.62iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 0.223T + 13T^{2} \)
17 \( 1 + 3.69T + 17T^{2} \)
23 \( 1 + 6.49iT - 23T^{2} \)
29 \( 1 - 6.57T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 8.71T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 - 2.95iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 1.82T + 73T^{2} \)
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

−8.797101312446899810861241068814, −8.248610021536731023210375520929, −7.40664522805829684594495614224, −6.77962087857234738656363934211, −6.38124772592024266158399393760, −5.05089604853988102654477194056, −3.81352528800858615301340932305, −2.59854784271875324489770152520, −1.42381966829404287055727586117, −0.46324710390559103497528091274, 2.48814582241437693558831417997, 3.16629192483419579908779619337, 4.33983404975611072330396256558, 5.15907698055409035076454770711, 5.63329201267292235156841779203, 6.70413436449514002925873041512, 8.359052510903860639203916533728, 8.891186162271538954552144852330, 9.280111750085338042551482640704, 10.15053444550387956077660807171

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