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Properties

Label 2-2448-17.13-c1-0-35
Degree $2$
Conductor $2448$
Sign $0.390 + 0.920i$
Analytic cond. $19.5473$
Root an. cond. $4.42124$
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 + (1.30 − 1.30i)7-s + (3.30 − 3.30i)11-s − 4.60·13-s + (3.60 + 2i)17-s − 6.60i·19-s + (−0.697 + 0.697i)23-s − 3i·25-s + (−5.60 − 5.60i)29-s + (−0.697 − 0.697i)31-s + 2.60·35-s + (3 + 3i)37-s + (−1 + i)41-s − 10.6i·43-s − 4·47-s + ⋯
L(s)  = 1  + (0.447 + 0.447i)5-s + (0.492 − 0.492i)7-s + (0.995 − 0.995i)11-s − 1.27·13-s + (0.874 + 0.485i)17-s − 1.51i·19-s + (−0.145 + 0.145i)23-s − 0.600i·25-s + (−1.04 − 1.04i)29-s + (−0.125 − 0.125i)31-s + 0.440·35-s + (0.493 + 0.493i)37-s + (−0.156 + 0.156i)41-s − 1.61i·43-s − 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.390 + 0.920i$
Analytic conductor: \(19.5473\)
Root analytic conductor: \(4.42124\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :1/2),\ 0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.893671022\)
\(L(\frac12)\) \(\approx\) \(1.893671022\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 1.30i)T - 7iT^{2} \)
11 \( 1 + (-3.30 + 3.30i)T - 11iT^{2} \)
13 \( 1 + 4.60T + 13T^{2} \)
19 \( 1 + 6.60iT - 19T^{2} \)
23 \( 1 + (0.697 - 0.697i)T - 23iT^{2} \)
29 \( 1 + (5.60 + 5.60i)T + 29iT^{2} \)
31 \( 1 + (0.697 + 0.697i)T + 31iT^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + (1 - i)T - 41iT^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 9.21iT - 53T^{2} \)
59 \( 1 - 1.39iT - 59T^{2} \)
61 \( 1 + (-8.21 + 8.21i)T - 61iT^{2} \)
67 \( 1 - 5.21T + 67T^{2} \)
71 \( 1 + (7.90 + 7.90i)T + 71iT^{2} \)
73 \( 1 + (7 + 7i)T + 73iT^{2} \)
79 \( 1 + (3.30 - 3.30i)T - 79iT^{2} \)
83 \( 1 - 3.81iT - 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + (0.394 + 0.394i)T + 97iT^{2} \)
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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

−8.884802217723981992189019530489, −7.928618519412791240985040939047, −7.29233972131941049790724891909, −6.47274798726065779907390527332, −5.76536745981095647955382831113, −4.81643905422845005015766510310, −3.95648195396128640185414777527, −2.96784595140194163612282618359, −1.97223583050255137732158870595, −0.64587685473476259250224405320, 1.41537597085258529158311580983, 2.09328422432092740951317891754, 3.39082907496971890488461991433, 4.41864932017395878705200408492, 5.23336789673958019630944758525, 5.75180161816307275016617271482, 6.91518835999444986408990611313, 7.51530805499297750987745361393, 8.333283738562182908093312650716, 9.255342891433625333879279717612

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