Properties

Label 2-1216-8.5-c1-0-5
Degree $2$
Conductor $1216$
Sign $-0.965 - 0.258i$
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  + i·3-s + 3.46i·5-s − 1.73·7-s + 2·9-s + 5.19i·13-s − 3.46·15-s − 3·17-s i·19-s − 1.73i·21-s + 1.73·23-s − 6.99·25-s + 5i·27-s + 1.73i·29-s + 3.46·31-s − 5.99i·35-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.54i·5-s − 0.654·7-s + 0.666·9-s + 1.44i·13-s − 0.894·15-s − 0.727·17-s − 0.229i·19-s − 0.377i·21-s + 0.361·23-s − 1.39·25-s + 0.962i·27-s + 0.321i·29-s + 0.622·31-s − 1.01i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.965 - 0.258i$
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: \((2,\ 1216,\ (\ :1/2),\ -0.965 - 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.219309721\)
\(L(\frac12)\) \(\approx\) \(1.219309721\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 3T^{2} \)
5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 + 1.73T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
23 \( 1 - 1.73T + 23T^{2} \)
29 \( 1 - 1.73iT - 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + 5.19iT - 53T^{2} \)
59 \( 1 - 9iT - 59T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 + 13iT - 67T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−10.08169268783341038303218263877, −9.513065209783156587392504017743, −8.677923727172064905131000703439, −7.27244607404216564447681596460, −6.83927656353014488929483641243, −6.22929062557113963510343636474, −4.81557542890447627150786534131, −3.92553215443835919497526020369, −3.10761673108875031430359833063, −1.99570936840885491694436235401, 0.52327539071794820116424229984, 1.57573625160565060131243997498, 3.01868337391652168154813455937, 4.29510467888817761876833666905, 5.04860448240018613682534738912, 6.01880978994877518340702791784, 6.84619942592678888942919242058, 7.993669900200742645136310306120, 8.305242105736197538967546392191, 9.464748671441932409039733944565

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