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Properties

Label 2-980-7.4-c3-0-36
Degree $2$
Conductor $980$
Sign $-0.701 + 0.712i$
Analytic cond. $57.8218$
Root an. cond. $7.60406$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 3.46i)3-s + (2.5 + 4.33i)5-s + (5.50 + 9.52i)9-s + (30 − 51.9i)11-s − 86·13-s + 20·15-s + (9 − 15.5i)17-s + (22 + 38.1i)19-s + (−24 − 41.5i)23-s + (−12.5 + 21.6i)25-s + 152·27-s − 186·29-s + (88 − 152. i)31-s + (−120 − 207. i)33-s + (−127 − 219. i)37-s + ⋯
L(s)  = 1  + (0.384 − 0.666i)3-s + (0.223 + 0.387i)5-s + (0.203 + 0.352i)9-s + (0.822 − 1.42i)11-s − 1.83·13-s + 0.344·15-s + (0.128 − 0.222i)17-s + (0.265 + 0.460i)19-s + (−0.217 − 0.376i)23-s + (−0.100 + 0.173i)25-s + 1.08·27-s − 1.19·29-s + (0.509 − 0.883i)31-s + (−0.633 − 1.09i)33-s + (−0.564 − 0.977i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(57.8218\)
Root analytic conductor: \(7.60406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :3/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.548948953\)
\(L(\frac12)\) \(\approx\) \(1.548948953\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 \)
good3 \( 1 + (-2 + 3.46i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (-30 + 51.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 86T + 2.19e3T^{2} \)
17 \( 1 + (-9 + 15.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-22 - 38.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (24 + 41.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 186T + 2.43e4T^{2} \)
31 \( 1 + (-88 + 152. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (127 + 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 186T + 6.89e4T^{2} \)
43 \( 1 + 100T + 7.95e4T^{2} \)
47 \( 1 + (-84 - 145. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-249 + 431. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (126 - 218. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (29 + 50.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-518 + 897. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 168T + 3.57e5T^{2} \)
73 \( 1 + (-253 + 438. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (136 + 235. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 948T + 5.71e5T^{2} \)
89 \( 1 + (507 + 878. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 766T + 9.12e5T^{2} \)
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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

−9.325244976210079359593563721425, −8.341565446525374288992805923732, −7.57429498570608488850016446446, −6.91910388234299659935407845349, −5.96562870077476561767770474943, −5.01593860549979173387449151354, −3.73499535039986700903981610021, −2.67583488708000378409213235899, −1.77747387553424690364463207866, −0.35136011162668215156990800766, 1.40947913267018933548670598600, 2.56108399182243948839960056833, 3.81809269215653066383738330233, 4.63267935103522204296159263675, 5.30465730930715664423313929007, 6.80051036420281735703027919894, 7.24396243222336478691446717079, 8.456505242091584106074258470348, 9.414837248999068069032615846532, 9.729286940912639186288375549760

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