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Properties

Label 2-460-115.68-c1-0-4
Degree $2$
Conductor $460$
Sign $-0.196 - 0.980i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (2.03 + 2.03i)3-s + (−1.06 + 1.96i)5-s + (0.641 + 0.641i)7-s + 5.28i·9-s − 3.68i·11-s + (1.51 + 1.51i)13-s + (−6.17 + 1.82i)15-s + (0.140 + 0.140i)17-s − 1.03·19-s + 2.61i·21-s + (−1.51 + 4.55i)23-s + (−2.71 − 4.19i)25-s + (−4.66 + 4.66i)27-s − 4.36i·29-s + 3.07·31-s + ⋯
L(s)  = 1  + (1.17 + 1.17i)3-s + (−0.477 + 0.878i)5-s + (0.242 + 0.242i)7-s + 1.76i·9-s − 1.11i·11-s + (0.420 + 0.420i)13-s + (−1.59 + 0.470i)15-s + (0.0341 + 0.0341i)17-s − 0.237·19-s + 0.569i·21-s + (−0.316 + 0.948i)23-s + (−0.543 − 0.839i)25-s + (−0.896 + 0.896i)27-s − 0.809i·29-s + 0.551·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.196 - 0.980i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.196 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20323 + 1.46783i\)
\(L(\frac12)\) \(\approx\) \(1.20323 + 1.46783i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.06 - 1.96i)T \)
23 \( 1 + (1.51 - 4.55i)T \)
good3 \( 1 + (-2.03 - 2.03i)T + 3iT^{2} \)
7 \( 1 + (-0.641 - 0.641i)T + 7iT^{2} \)
11 \( 1 + 3.68iT - 11T^{2} \)
13 \( 1 + (-1.51 - 1.51i)T + 13iT^{2} \)
17 \( 1 + (-0.140 - 0.140i)T + 17iT^{2} \)
19 \( 1 + 1.03T + 19T^{2} \)
29 \( 1 + 4.36iT - 29T^{2} \)
31 \( 1 - 3.07T + 31T^{2} \)
37 \( 1 + (-3.28 - 3.28i)T + 37iT^{2} \)
41 \( 1 + 8.10T + 41T^{2} \)
43 \( 1 + (-4.71 + 4.71i)T - 43iT^{2} \)
47 \( 1 + (1.77 - 1.77i)T - 47iT^{2} \)
53 \( 1 + (-10.0 + 10.0i)T - 53iT^{2} \)
59 \( 1 + 6.32iT - 59T^{2} \)
61 \( 1 - 13.9iT - 61T^{2} \)
67 \( 1 + (-7.97 - 7.97i)T + 67iT^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + (-0.0704 - 0.0704i)T + 73iT^{2} \)
79 \( 1 - 9.66T + 79T^{2} \)
83 \( 1 + (5.33 - 5.33i)T - 83iT^{2} \)
89 \( 1 - 8.40T + 89T^{2} \)
97 \( 1 + (7.74 + 7.74i)T + 97iT^{2} \)
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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

−11.18561303614017205352926655285, −10.29491084326658016034182434287, −9.573077234802585207692010168774, −8.465340846357963673550035228466, −8.127220319288300690924363749283, −6.77842853982673924139181509618, −5.50784548982296043888745775442, −4.09924415625731876904177662413, −3.48104270707641723460148816490, −2.43134941349634977902489906591, 1.15383506594464846397472599919, 2.36832643194683183477379406907, 3.77441992303862742892447175730, 4.88125870462654788607128507455, 6.43505260496533827016692518068, 7.42383387123845158273598528669, 8.043841074000849847497160328193, 8.738341091232016377918393801689, 9.599084906071959457789539133028, 10.84775074399291326108090845077

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