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Properties

Label 2-612-17.4-c1-0-3
Degree $2$
Conductor $612$
Sign $0.987 - 0.155i$
Analytic cond. $4.88684$
Root an. cond. $2.21062$
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  + (1 − i)5-s + (2.30 + 2.30i)7-s + (0.302 + 0.302i)11-s + 2.60·13-s + (−3.60 − 2i)17-s + 0.605i·19-s + (4.30 + 4.30i)23-s + 3i·25-s + (1.60 − 1.60i)29-s + (4.30 − 4.30i)31-s + 4.60·35-s + (3 − 3i)37-s + (−1 − i)41-s − 3.39i·43-s + 4·47-s + ⋯
L(s)  = 1  + (0.447 − 0.447i)5-s + (0.870 + 0.870i)7-s + (0.0912 + 0.0912i)11-s + 0.722·13-s + (−0.874 − 0.485i)17-s + 0.138i·19-s + (0.897 + 0.897i)23-s + 0.600i·25-s + (0.298 − 0.298i)29-s + (0.772 − 0.772i)31-s + 0.778·35-s + (0.493 − 0.493i)37-s + (−0.156 − 0.156i)41-s − 0.517i·43-s + 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(612\)    =    \(2^{2} \cdot 3^{2} \cdot 17\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(4.88684\)
Root analytic conductor: \(2.21062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{612} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 612,\ (\ :1/2),\ 0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77787 + 0.139370i\)
\(L(\frac12)\) \(\approx\) \(1.77787 + 0.139370i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 + (3.60 + 2i)T \)
good5 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 + (-2.30 - 2.30i)T + 7iT^{2} \)
11 \( 1 + (-0.302 - 0.302i)T + 11iT^{2} \)
13 \( 1 - 2.60T + 13T^{2} \)
19 \( 1 - 0.605iT - 19T^{2} \)
23 \( 1 + (-4.30 - 4.30i)T + 23iT^{2} \)
29 \( 1 + (-1.60 + 1.60i)T - 29iT^{2} \)
31 \( 1 + (-4.30 + 4.30i)T - 31iT^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + (1 + i)T + 41iT^{2} \)
43 \( 1 + 3.39iT - 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 5.21iT - 53T^{2} \)
59 \( 1 - 8.60iT - 59T^{2} \)
61 \( 1 + (6.21 + 6.21i)T + 61iT^{2} \)
67 \( 1 - 9.21T + 67T^{2} \)
71 \( 1 + (2.90 - 2.90i)T - 71iT^{2} \)
73 \( 1 + (7 - 7i)T - 73iT^{2} \)
79 \( 1 + (0.302 + 0.302i)T + 79iT^{2} \)
83 \( 1 + 17.8iT - 83T^{2} \)
89 \( 1 + 7.81T + 89T^{2} \)
97 \( 1 + (7.60 - 7.60i)T - 97iT^{2} \)
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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

−10.83379034188173966249810306611, −9.545217052836924923207619416191, −8.960086159328739392482155373006, −8.203781922414914358028657227078, −7.12436865859873432008694555095, −5.92089277419119084451708508343, −5.23446268886370302180408312502, −4.20796097066910155523393958465, −2.64244456102297184060446566906, −1.43627071674702977818005897393, 1.26823937978093636537366020667, 2.71164280887768087837494649113, 4.10041197291878042632738966249, 4.94006189918619459662354571954, 6.31587785112929278359369857293, 6.88098979035635875136483210398, 8.094089480722357763469376659316, 8.721903032845293669599431539288, 9.919892769164562341856672547650, 10.82614247478464190530043154396

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