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Properties

Label 2-650-65.64-c1-0-9
Degree $2$
Conductor $650$
Sign $0.534 + 0.845i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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-s − 0.223i·3-s + 4-s + 0.223i·6-s + 0.776·7-s − 8-s + 2.95·9-s − 4.72i·11-s − 0.223i·12-s + (−3.08 − 1.86i)13-s − 0.776·14-s + 16-s + 3i·17-s − 2.95·18-s + 2.22i·19-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.129i·3-s + 0.5·4-s + 0.0912i·6-s + 0.293·7-s − 0.353·8-s + 0.983·9-s − 1.42i·11-s − 0.0645i·12-s + (−0.856 − 0.516i)13-s − 0.207·14-s + 0.250·16-s + 0.727i·17-s − 0.695·18-s + 0.510i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.534 + 0.845i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.534 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.963842 - 0.530775i\)
\(L(\frac12)\) \(\approx\) \(0.963842 - 0.530775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
13 \( 1 + (3.08 + 1.86i)T \)
good3 \( 1 + 0.223iT - 3T^{2} \)
7 \( 1 - 0.776T + 7T^{2} \)
11 \( 1 + 4.72iT - 11T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 - 2.22iT - 19T^{2} \)
23 \( 1 + 6.17iT - 23T^{2} \)
29 \( 1 - 8.50T + 29T^{2} \)
31 \( 1 - 0.950iT - 31T^{2} \)
37 \( 1 - 1.72T + 37T^{2} \)
41 \( 1 + 8.39iT - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 5.39T + 47T^{2} \)
53 \( 1 + 2.50iT - 53T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 + 4.32T + 61T^{2} \)
67 \( 1 - 9.17T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 7.05T + 73T^{2} \)
79 \( 1 - 0.273T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−10.46663138136892812346779240201, −9.620007789839580705592781569727, −8.457083934335540728024336985073, −8.058212922089586176857287725300, −6.94231218992279150631768235507, −6.13296089768353663392130889476, −4.95570111858144732888042583715, −3.65376381824739563757526369469, −2.31967106478693151756859048068, −0.825140922864451005014035256564, 1.45837077689139638924052817250, 2.65434586748638500911510422881, 4.34815489856845310033680268657, 5.01335444476402002018519592753, 6.61237827305999091336220793051, 7.27289639607529747875852566927, 7.923279252987914024578116923948, 9.334261752372451969558527514806, 9.656377497734655989518903836793, 10.42949287410711621094667224899

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