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Properties

Label 2-980-7.4-c1-0-13
Degree $2$
Conductor $980$
Sign $-0.947 + 0.318i$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
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  + (1.20 − 2.09i)3-s + (−0.5 − 0.866i)5-s + (−1.41 − 2.44i)9-s + (−0.914 + 1.58i)11-s − 6.41·13-s − 2.41·15-s + (1.79 − 3.10i)17-s + (−3.82 − 6.63i)19-s + (−1.70 − 2.95i)23-s + (−0.499 + 0.866i)25-s + 0.414·27-s − 4.65·29-s + (3.70 − 6.42i)31-s + (2.20 + 3.82i)33-s + (0.292 + 0.507i)37-s + ⋯
L(s)  = 1  + (0.696 − 1.20i)3-s + (−0.223 − 0.387i)5-s + (−0.471 − 0.816i)9-s + (−0.275 + 0.477i)11-s − 1.77·13-s − 0.623·15-s + (0.434 − 0.753i)17-s + (−0.878 − 1.52i)19-s + (−0.355 − 0.616i)23-s + (−0.0999 + 0.173i)25-s + 0.0797·27-s − 0.864·29-s + (0.665 − 1.15i)31-s + (0.384 + 0.665i)33-s + (0.0481 + 0.0834i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.947 + 0.318i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.203717 - 1.24562i\)
\(L(\frac12)\) \(\approx\) \(0.203717 - 1.24562i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-1.20 + 2.09i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.914 - 1.58i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.41T + 13T^{2} \)
17 \( 1 + (-1.79 + 3.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.82 + 6.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.70 + 2.95i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.65T + 29T^{2} \)
31 \( 1 + (-3.70 + 6.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.292 - 0.507i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
43 \( 1 - 0.343T + 43T^{2} \)
47 \( 1 + (-5.44 - 9.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.12 + 10.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.292 + 0.507i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.41 - 9.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.53 - 2.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + (-5.41 + 9.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.57 - 13.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (8.48 + 14.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.72T + 97T^{2} \)
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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

−9.459504147198602458678462898766, −8.697223130511795870088847679661, −7.72002749068609437113353631564, −7.34298444866798136313455882616, −6.53603596718866712310649130692, −5.16538428132582563149264536619, −4.37209765315206811992515655445, −2.70201144718401015257576039169, −2.18692360294010588135582098552, −0.49710542334889979006308684070, 2.16953722663812002115986305904, 3.31436461484153477365915187222, 3.97832525418577425419372863642, 4.99102246117350731149441231988, 5.93211686092984292947494401388, 7.19614249230403352347077863927, 8.045083344450817300347096402220, 8.734232192970022528439096956454, 9.764894745715743559680888362564, 10.21175554962225006527425178063

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