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Properties

Label 2-980-35.4-c1-0-3
Degree $2$
Conductor $980$
Sign $0.946 - 0.324i$
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  + (−2.59 − 1.5i)3-s + (−0.133 − 2.23i)5-s + (3 + 5.19i)9-s + (−1.5 + 2.59i)11-s + i·13-s + (−3 + 6i)15-s + (−4.33 − 2.5i)17-s + (4 + 6.92i)19-s + (−1.73 + i)23-s + (−4.96 + 0.598i)25-s − 9i·27-s + 29-s + (−1 + 1.73i)31-s + (7.79 − 4.5i)33-s + (8.66 − 5i)37-s + ⋯
L(s)  = 1  + (−1.49 − 0.866i)3-s + (−0.0599 − 0.998i)5-s + (1 + 1.73i)9-s + (−0.452 + 0.783i)11-s + 0.277i·13-s + (−0.774 + 1.54i)15-s + (−1.05 − 0.606i)17-s + (0.917 + 1.58i)19-s + (−0.361 + 0.208i)23-s + (−0.992 + 0.119i)25-s − 1.73i·27-s + 0.185·29-s + (−0.179 + 0.311i)31-s + (1.35 − 0.783i)33-s + (1.42 − 0.821i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.946 - 0.324i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.946 - 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.604357 + 0.100649i\)
\(L(\frac12)\) \(\approx\) \(0.604357 + 0.100649i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.133 + 2.23i)T \)
7 \( 1 \)
good3 \( 1 + (2.59 + 1.5i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + (4.33 + 2.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 - i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.66 + 5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (9.52 - 5.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.66 - 5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-8.66 - 5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + (4 + 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3iT - 97T^{2} \)
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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

−10.04984564198975422885161118723, −9.390437329431995130681000048556, −8.060908302065309439457015467889, −7.50332231487858494944087204237, −6.54246563685918802109861910950, −5.69432002156047575343929288719, −5.02270565418853954589469780708, −4.17458753199840679756951925377, −2.10716521083837070371705689821, −1.02484058119904222543029692529, 0.42636961076067710366381340074, 2.67915935985273201602604406358, 3.82845209086311144048413634782, 4.78820270560852647865282149702, 5.67800944691663403298356849737, 6.38811216024287723597101720380, 7.10365631757318600519899018152, 8.317371218635312220296071027912, 9.489162150510769453158272970996, 10.15665497531042525440201299405

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