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Properties

Label 2-980-980.659-c0-0-1
Degree $2$
Conductor $980$
Sign $0.926 - 0.375i$
Analytic cond. $0.489083$
Root an. cond. $0.699345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.900 − 0.433i)2-s + (−0.0990 + 0.433i)3-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + (0.0990 + 0.433i)6-s + (−0.623 + 0.781i)7-s + (0.222 − 0.974i)8-s + (0.722 + 0.347i)9-s + (0.222 + 0.974i)10-s + (0.277 + 0.347i)12-s + (−0.222 + 0.974i)14-s + (−0.400 − 0.193i)15-s + (−0.222 − 0.974i)16-s + 0.801·18-s + (0.623 + 0.781i)20-s + (−0.277 − 0.347i)21-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (−0.0990 + 0.433i)3-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + (0.0990 + 0.433i)6-s + (−0.623 + 0.781i)7-s + (0.222 − 0.974i)8-s + (0.722 + 0.347i)9-s + (0.222 + 0.974i)10-s + (0.277 + 0.347i)12-s + (−0.222 + 0.974i)14-s + (−0.400 − 0.193i)15-s + (−0.222 − 0.974i)16-s + 0.801·18-s + (0.623 + 0.781i)20-s + (−0.277 − 0.347i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.926 - 0.375i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.926 - 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.570117256\)
\(L(\frac12)\) \(\approx\) \(1.570117256\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (0.623 - 0.781i)T \)
good3 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.623 + 0.781i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.222 - 0.974i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.222 - 0.974i)T^{2} \)
41 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.222 + 0.974i)T^{2} \)
59 \( 1 + (0.900 - 0.433i)T^{2} \)
61 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.623 - 0.781i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.47295045395071605995503339335, −9.776879191469905304583624408570, −8.804903886402440053245749306418, −7.35586770088120817817760853387, −6.75006782821100620190893003951, −5.83631361524949325386797594191, −4.94452096881994648187353747090, −3.88487061794276791787058824307, −3.07531136508925290234514039807, −2.06290134824308518935578865699, 1.36149034387707679416999370316, 3.06655090726072094378133430761, 4.18077976862656050492643992791, 4.70300686385153167233292855979, 5.99048926940434547225617304879, 6.62932266731055931206672663095, 7.55893210346583017977231903740, 8.106387346385413306675734708847, 9.310326683060464550911415611762, 10.09271785458841409102823777714

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