Properties

Label 2-1216-76.75-c1-0-31
Degree $2$
Conductor $1216$
Sign $0.844 + 0.535i$
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.35·3-s + 2.56·5-s − 4.15i·7-s + 2.56·9-s − 2.33i·11-s + 4.29i·13-s + 6.04·15-s − 17-s + (3.68 + 2.33i)19-s − 9.79i·21-s + 1.82i·23-s + 1.56·25-s − 1.03·27-s + 1.20i·29-s + 1.32·31-s + ⋯
L(s)  = 1  + 1.36·3-s + 1.14·5-s − 1.56i·7-s + 0.853·9-s − 0.703i·11-s + 1.19i·13-s + 1.55·15-s − 0.242·17-s + (0.844 + 0.535i)19-s − 2.13i·21-s + 0.379i·23-s + 0.312·25-s − 0.198·27-s + 0.223i·29-s + 0.237·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.844 + 0.535i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.844 + 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.146784414\)
\(L(\frac12)\) \(\approx\) \(3.146784414\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.68 - 2.33i)T \)
good3 \( 1 - 2.35T + 3T^{2} \)
5 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 + 4.15iT - 7T^{2} \)
11 \( 1 + 2.33iT - 11T^{2} \)
13 \( 1 - 4.29iT - 13T^{2} \)
17 \( 1 + T + 17T^{2} \)
23 \( 1 - 1.82iT - 23T^{2} \)
29 \( 1 - 1.20iT - 29T^{2} \)
31 \( 1 - 1.32T + 31T^{2} \)
37 \( 1 + 5.49iT - 37T^{2} \)
41 \( 1 + 5.49iT - 41T^{2} \)
43 \( 1 - 1.30iT - 43T^{2} \)
47 \( 1 + 6.99iT - 47T^{2} \)
53 \( 1 - 9.79iT - 53T^{2} \)
59 \( 1 - 6.33T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + 0.290T + 67T^{2} \)
71 \( 1 + 2.06T + 71T^{2} \)
73 \( 1 + 0.123T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 - 14.0iT - 89T^{2} \)
97 \( 1 - 2.41iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.440279578336070835583130334183, −9.081738234203564075248299976513, −8.019188794819516002596355494790, −7.31758327359972982643036746338, −6.50837048545411335472581259291, −5.44218725535420681900103520224, −4.12929586865785206250528241022, −3.50190986633198472985929588709, −2.31010804065093342620906739905, −1.31255685127914307311767126837, 1.77757782025173532871210799920, 2.60327592727165964909293369584, 3.14054315114138498032335974684, 4.77867085699922091620480788561, 5.60014759285111154186900707785, 6.38262871508927122819468859088, 7.60741912083150485041624600494, 8.353119420695452026164942014469, 9.035948889058838956918688761930, 9.633735363911708196217766214153

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