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Properties

Label 2-416-52.31-c1-0-11
Degree $2$
Conductor $416$
Sign $-0.289 + 0.957i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
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  − 2i·3-s + (−1 + i)5-s + (1 − i)7-s − 9-s + (3 − 3i)11-s + (−3 − 2i)13-s + (2 + 2i)15-s − 4i·17-s + (−3 − 3i)19-s + (−2 − 2i)21-s + 3i·25-s − 4i·27-s − 6·29-s + (3 + 3i)31-s + (−6 − 6i)33-s + ⋯
L(s)  = 1  − 1.15i·3-s + (−0.447 + 0.447i)5-s + (0.377 − 0.377i)7-s − 0.333·9-s + (0.904 − 0.904i)11-s + (−0.832 − 0.554i)13-s + (0.516 + 0.516i)15-s − 0.970i·17-s + (−0.688 − 0.688i)19-s + (−0.436 − 0.436i)21-s + 0.600i·25-s − 0.769i·27-s − 1.11·29-s + (0.538 + 0.538i)31-s + (−1.04 − 1.04i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.289 + 0.957i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.289 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.742334 - 1.00037i\)
\(L(\frac12)\) \(\approx\) \(0.742334 - 1.00037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (3 + 2i)T \)
good3 \( 1 + 2iT - 3T^{2} \)
5 \( 1 + (1 - i)T - 5iT^{2} \)
7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 + (-3 + 3i)T - 11iT^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-3 - 3i)T + 31iT^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + (-1 + i)T - 41iT^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-5 + 5i)T - 47iT^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-7 + 7i)T - 59iT^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + (-5 - 5i)T + 67iT^{2} \)
71 \( 1 + (5 + 5i)T + 71iT^{2} \)
73 \( 1 + (-9 - 9i)T + 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (7 + 7i)T + 83iT^{2} \)
89 \( 1 + (-5 - 5i)T + 89iT^{2} \)
97 \( 1 + (-13 + 13i)T - 97iT^{2} \)
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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

−11.31392185273489285356462307776, −10.12777678751927162915931077554, −8.944229712393368449988817767770, −7.933929834186261852628887955234, −7.17275869981612183839742504189, −6.57592182355326090814732601396, −5.22629001695797034811649077698, −3.82050309870889071469739403267, −2.47536426987382414381381696534, −0.846320689843376080608163233163, 2.00355756099595408409347565738, 4.02296685213288732680880407294, 4.32118410482290499775206239259, 5.49226295140879080486505428328, 6.80045863406213119772349186885, 7.990392797744011416122359405847, 8.933277050891136879932285194976, 9.689401752764033736154683050459, 10.40985395938440876846530180153, 11.50949877339205199146872425380

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