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Properties

Label 2-416-52.31-c1-0-6
Degree $2$
Conductor $416$
Sign $0.984 - 0.176i$
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  − i·3-s + (−1.58 + 1.58i)5-s + (−1.58 + 1.58i)7-s + 2·9-s + (4.16 − 4.16i)11-s + (3.58 + 0.418i)13-s + (1.58 + 1.58i)15-s + 7.32i·17-s + (1.16 + 1.16i)19-s + (1.58 + 1.58i)21-s + 7.16·23-s − 5i·27-s − 1.16·29-s + (−1.16 − 1.16i)31-s + (−4.16 − 4.16i)33-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.707 + 0.707i)5-s + (−0.597 + 0.597i)7-s + 0.666·9-s + (1.25 − 1.25i)11-s + (0.993 + 0.116i)13-s + (0.408 + 0.408i)15-s + 1.77i·17-s + (0.266 + 0.266i)19-s + (0.345 + 0.345i)21-s + 1.49·23-s − 0.962i·27-s − 0.215·29-s + (−0.208 − 0.208i)31-s + (−0.724 − 0.724i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.984 - 0.176i$
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: \((2,\ 416,\ (\ :1/2),\ 0.984 - 0.176i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32953 + 0.118351i\)
\(L(\frac12)\) \(\approx\) \(1.32953 + 0.118351i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.58 - 0.418i)T \)
good3 \( 1 + iT - 3T^{2} \)
5 \( 1 + (1.58 - 1.58i)T - 5iT^{2} \)
7 \( 1 + (1.58 - 1.58i)T - 7iT^{2} \)
11 \( 1 + (-4.16 + 4.16i)T - 11iT^{2} \)
17 \( 1 - 7.32iT - 17T^{2} \)
19 \( 1 + (-1.16 - 1.16i)T + 19iT^{2} \)
23 \( 1 - 7.16T + 23T^{2} \)
29 \( 1 + 1.16T + 29T^{2} \)
31 \( 1 + (1.16 + 1.16i)T + 31iT^{2} \)
37 \( 1 + (3.58 + 3.58i)T + 37iT^{2} \)
41 \( 1 + (-5.16 + 5.16i)T - 41iT^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (6.74 - 6.74i)T - 47iT^{2} \)
53 \( 1 - 9.48T + 53T^{2} \)
59 \( 1 + (4 - 4i)T - 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (7.32 + 7.32i)T + 67iT^{2} \)
71 \( 1 + (1.58 + 1.58i)T + 71iT^{2} \)
73 \( 1 + (6 + 6i)T + 73iT^{2} \)
79 \( 1 + 3.48iT - 79T^{2} \)
83 \( 1 + (-5.83 - 5.83i)T + 83iT^{2} \)
89 \( 1 + (2.83 + 2.83i)T + 89iT^{2} \)
97 \( 1 + (3.83 - 3.83i)T - 97iT^{2} \)
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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

−11.21099385668825825741397662566, −10.62139598513155544565657799330, −9.208069948101888539678208352365, −8.527541852753756860544006291176, −7.42451545023285964706178686726, −6.47447744590984896716212535394, −5.93501140011661582698058221313, −3.93994847228992022849843275917, −3.28760533636346068304922633387, −1.40064698712683332706225680158, 1.11482002652163050123247441336, 3.39044324178106615893187933425, 4.31733002795279148427775541180, 5.00075067304222311909214820900, 6.83132490313818765493130250903, 7.21494608279508074477071064344, 8.688137590907587223340453864870, 9.445178302844407111346128072562, 10.07718615347605479207250082621, 11.28330529111082331356784376077

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