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Properties

Label 2-460-5.4-c1-0-1
Degree $2$
Conductor $460$
Sign $-0.930 - 0.365i$
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  + 2.40i·3-s + (−0.817 + 2.08i)5-s + 4.41i·7-s − 2.76·9-s + 2.29·11-s − 6.92i·13-s + (−4.99 − 1.96i)15-s + 1.51i·17-s − 2.89·19-s − 10.5·21-s − i·23-s + (−3.66 − 3.40i)25-s + 0.570i·27-s + 7.68·29-s + 3.85·31-s + ⋯
L(s)  = 1  + 1.38i·3-s + (−0.365 + 0.930i)5-s + 1.66i·7-s − 0.920·9-s + 0.691·11-s − 1.92i·13-s + (−1.29 − 0.506i)15-s + 0.367i·17-s − 0.665·19-s − 2.31·21-s − 0.208i·23-s + (−0.732 − 0.680i)25-s + 0.109i·27-s + 1.42·29-s + 0.692·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.930 - 0.365i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.930 - 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.229094 + 1.21063i\)
\(L(\frac12)\) \(\approx\) \(0.229094 + 1.21063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.817 - 2.08i)T \)
23 \( 1 + iT \)
good3 \( 1 - 2.40iT - 3T^{2} \)
7 \( 1 - 4.41iT - 7T^{2} \)
11 \( 1 - 2.29T + 11T^{2} \)
13 \( 1 + 6.92iT - 13T^{2} \)
17 \( 1 - 1.51iT - 17T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
29 \( 1 - 7.68T + 29T^{2} \)
31 \( 1 - 3.85T + 31T^{2} \)
37 \( 1 - 8.62iT - 37T^{2} \)
41 \( 1 + 6.44T + 41T^{2} \)
43 \( 1 - 3.48iT - 43T^{2} \)
47 \( 1 + 6.19iT - 47T^{2} \)
53 \( 1 + 2.17iT - 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 5.11T + 61T^{2} \)
67 \( 1 - 9.94iT - 67T^{2} \)
71 \( 1 - 3.41T + 71T^{2} \)
73 \( 1 - 8.95iT - 73T^{2} \)
79 \( 1 - 1.92T + 79T^{2} \)
83 \( 1 + 8.04iT - 83T^{2} \)
89 \( 1 - 1.09T + 89T^{2} \)
97 \( 1 + 16.9iT - 97T^{2} \)
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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

−11.37784252795063039684689942122, −10.22611531721034532851730491567, −10.06740285625187454217321327607, −8.675032345791187376637395539322, −8.242737367927387372909335540009, −6.58700795934489355472324400352, −5.70618625533653052266543781331, −4.72889976112777422724668390684, −3.44115399700789175677946960151, −2.68652198400340543076526739409, 0.807521229767803775095266743641, 1.81670284360174104369425201412, 3.93556561785743818778666533754, 4.63331926001350761106780700083, 6.44250514437075508429597942728, 6.94462854997578601715277197945, 7.75274036243254929524072743302, 8.687987540221761081924645614826, 9.620307917883084293794094109453, 10.88409025841698253886799551730

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