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Properties

Label 2-416-52.47-c1-0-9
Degree $2$
Conductor $416$
Sign $-0.238 + 0.971i$
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  + 2.89i·3-s + (−2.84 − 2.84i)5-s + (0.193 + 0.193i)7-s − 5.40·9-s + (−3.65 − 3.65i)11-s + (−0.193 − 3.60i)13-s + (8.25 − 8.25i)15-s + 0.899i·17-s + (−0.753 + 0.753i)19-s + (−0.560 + 0.560i)21-s + 1.89·23-s + 11.2i·25-s − 6.97i·27-s − 5.41·29-s + (−3.24 + 3.24i)31-s + ⋯
L(s)  = 1  + 1.67i·3-s + (−1.27 − 1.27i)5-s + (0.0730 + 0.0730i)7-s − 1.80·9-s + (−1.10 − 1.10i)11-s + (−0.0536 − 0.998i)13-s + (2.13 − 2.13i)15-s + 0.218i·17-s + (−0.172 + 0.172i)19-s + (−0.122 + 0.122i)21-s + 0.394·23-s + 2.24i·25-s − 1.34i·27-s − 1.00·29-s + (−0.583 + 0.583i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.238 + 0.971i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.238 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.199453 - 0.254242i\)
\(L(\frac12)\) \(\approx\) \(0.199453 - 0.254242i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.193 + 3.60i)T \)
good3 \( 1 - 2.89iT - 3T^{2} \)
5 \( 1 + (2.84 + 2.84i)T + 5iT^{2} \)
7 \( 1 + (-0.193 - 0.193i)T + 7iT^{2} \)
11 \( 1 + (3.65 + 3.65i)T + 11iT^{2} \)
17 \( 1 - 0.899iT - 17T^{2} \)
19 \( 1 + (0.753 - 0.753i)T - 19iT^{2} \)
23 \( 1 - 1.89T + 23T^{2} \)
29 \( 1 + 5.41T + 29T^{2} \)
31 \( 1 + (3.24 - 3.24i)T - 31iT^{2} \)
37 \( 1 + (-2.95 + 2.95i)T - 37iT^{2} \)
41 \( 1 + (5.79 + 5.79i)T + 41iT^{2} \)
43 \( 1 - 2.81T + 43T^{2} \)
47 \( 1 + (7.49 + 7.49i)T + 47iT^{2} \)
53 \( 1 + 5.69T + 53T^{2} \)
59 \( 1 + (-8.44 - 8.44i)T + 59iT^{2} \)
61 \( 1 + 2.98T + 61T^{2} \)
67 \( 1 + (8.85 - 8.85i)T - 67iT^{2} \)
71 \( 1 + (-1.91 + 1.91i)T - 71iT^{2} \)
73 \( 1 + (-5.40 + 5.40i)T - 73iT^{2} \)
79 \( 1 - 1.20iT - 79T^{2} \)
83 \( 1 + (2.14 - 2.14i)T - 83iT^{2} \)
89 \( 1 + (1.79 - 1.79i)T - 89iT^{2} \)
97 \( 1 + (1.10 + 1.10i)T + 97iT^{2} \)
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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

−10.89680714623827368587308091041, −10.16404102340475703818902470081, −8.948020250921431013571089678016, −8.482932488754743076400649036467, −7.66591706689284879051867025934, −5.48038372022133356088541301311, −5.14527009888757773551731458284, −3.97247688082238090356795433045, −3.25596987959983201993680493349, −0.19844666344780292348829553447, 2.02847603382131145394289962338, 3.07724783659772894225414586674, 4.59222830101812946284472722742, 6.26934109910490637154373005614, 7.08648020148768209942581341428, 7.52939440379462712043503444801, 8.192169771777283976115947614111, 9.697343510933547461504649355425, 11.07787048291792561043461480040, 11.40895679432147528833670385298

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