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Properties

Label 2-416-52.31-c1-0-4
Degree $2$
Conductor $416$
Sign $-0.797 - 0.602i$
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  + 3.16i·3-s + (1.68 − 1.68i)5-s + (−2.12 + 2.12i)7-s − 7.03·9-s + (−1.43 + 1.43i)11-s + (2.12 + 2.91i)13-s + (5.34 + 5.34i)15-s + 5.16i·17-s + (−4.60 − 4.60i)19-s + (−6.72 − 6.72i)21-s + 4.95·23-s − 0.700i·25-s − 12.7i·27-s + 2.09·29-s + (0.602 + 0.602i)31-s + ⋯
L(s)  = 1  + 1.82i·3-s + (0.755 − 0.755i)5-s + (−0.802 + 0.802i)7-s − 2.34·9-s + (−0.432 + 0.432i)11-s + (0.588 + 0.808i)13-s + (1.38 + 1.38i)15-s + 1.25i·17-s + (−1.05 − 1.05i)19-s + (−1.46 − 1.46i)21-s + 1.03·23-s − 0.140i·25-s − 2.46i·27-s + 0.388·29-s + (0.108 + 0.108i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.797 - 0.602i$
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: \((2,\ 416,\ (\ :1/2),\ -0.797 - 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.384163 + 1.14532i\)
\(L(\frac12)\) \(\approx\) \(0.384163 + 1.14532i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 - 2.91i)T \)
good3 \( 1 - 3.16iT - 3T^{2} \)
5 \( 1 + (-1.68 + 1.68i)T - 5iT^{2} \)
7 \( 1 + (2.12 - 2.12i)T - 7iT^{2} \)
11 \( 1 + (1.43 - 1.43i)T - 11iT^{2} \)
17 \( 1 - 5.16iT - 17T^{2} \)
19 \( 1 + (4.60 + 4.60i)T + 19iT^{2} \)
23 \( 1 - 4.95T + 23T^{2} \)
29 \( 1 - 2.09T + 29T^{2} \)
31 \( 1 + (-0.602 - 0.602i)T + 31iT^{2} \)
37 \( 1 + (4.64 + 4.64i)T + 37iT^{2} \)
41 \( 1 + (2.79 - 2.79i)T - 41iT^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + (0.746 - 0.746i)T - 47iT^{2} \)
53 \( 1 - 3.37T + 53T^{2} \)
59 \( 1 + (-3.22 + 3.22i)T - 59iT^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 + (5.26 + 5.26i)T + 67iT^{2} \)
71 \( 1 + (-1.16 - 1.16i)T + 71iT^{2} \)
73 \( 1 + (-7.03 - 7.03i)T + 73iT^{2} \)
79 \( 1 - 0.171iT - 79T^{2} \)
83 \( 1 + (-7.77 - 7.77i)T + 83iT^{2} \)
89 \( 1 + (-1.20 - 1.20i)T + 89iT^{2} \)
97 \( 1 + (-1.95 + 1.95i)T - 97iT^{2} \)
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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

−11.22562088732631662107032806688, −10.52616910613154580619693796660, −9.646725230448179311885576149382, −8.987574552404927840864978094426, −8.612883132998142016940772044856, −6.50263464834587548324606933567, −5.59362975954392917897563001416, −4.78598023046951000527566948672, −3.79052376990841061178971309730, −2.42878868056236488565638141133, 0.76955246989288725567850298021, 2.38254959747504815388869080491, 3.29776651971823596887716456769, 5.54824629025947025530469597846, 6.40220975372659923442100304867, 6.95195270073837475520515581727, 7.83199331489499320106988350312, 8.801805561750726932433092735176, 10.19611159882262335184566454157, 10.80602788806939340485768744686

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