Properties

Label 2-980-140.107-c1-0-20
Degree $2$
Conductor $980$
Sign $0.800 - 0.599i$
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  + (0.479 − 1.33i)2-s + (0.298 − 1.11i)3-s + (−1.53 − 1.27i)4-s + (−0.308 + 2.21i)5-s + (−1.33 − 0.930i)6-s + (−2.43 + 1.43i)8-s + (1.44 + 0.837i)9-s + (2.79 + 1.47i)10-s + (−1.59 + 0.918i)11-s + (−1.87 + 1.33i)12-s + (−2.24 + 2.24i)13-s + (2.37 + 1.00i)15-s + (0.738 + 3.93i)16-s + (−1.27 + 4.75i)17-s + (1.80 − 1.52i)18-s + (−3.96 + 6.87i)19-s + ⋯
L(s)  = 1  + (0.339 − 0.940i)2-s + (0.172 − 0.642i)3-s + (−0.769 − 0.638i)4-s + (−0.137 + 0.990i)5-s + (−0.545 − 0.379i)6-s + (−0.861 + 0.507i)8-s + (0.483 + 0.279i)9-s + (0.884 + 0.465i)10-s + (−0.479 + 0.276i)11-s + (−0.542 + 0.384i)12-s + (−0.623 + 0.623i)13-s + (0.612 + 0.258i)15-s + (0.184 + 0.982i)16-s + (−0.308 + 1.15i)17-s + (0.426 − 0.359i)18-s + (−0.910 + 1.57i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.800 - 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00340 + 0.334324i\)
\(L(\frac12)\) \(\approx\) \(1.00340 + 0.334324i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.479 + 1.33i)T \)
5 \( 1 + (0.308 - 2.21i)T \)
7 \( 1 \)
good3 \( 1 + (-0.298 + 1.11i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.59 - 0.918i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.24 - 2.24i)T - 13iT^{2} \)
17 \( 1 + (1.27 - 4.75i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.96 - 6.87i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.12 - 1.37i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.07iT - 29T^{2} \)
31 \( 1 + (-3.36 + 1.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.389 - 0.104i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.61T + 41T^{2} \)
43 \( 1 + (5.11 + 5.11i)T + 43iT^{2} \)
47 \( 1 + (-2.87 - 10.7i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.65 - 0.978i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.83 + 4.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.16 + 2.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.24 - 1.40i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.56iT - 71T^{2} \)
73 \( 1 + (4.59 + 1.23i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.67 + 9.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.898 + 0.898i)T + 83iT^{2} \)
89 \( 1 + (-13.2 - 7.64i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.82 - 8.82i)T + 97iT^{2} \)
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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

−10.26566212856852348288512857595, −9.657941991041947722532702070268, −8.240059195380908956155753100932, −7.74445456599057349576296389967, −6.53928552143932846812607782716, −5.90851929524359672527013818762, −4.43475990265879302944271211375, −3.77774008012547496848232652485, −2.35054415092149351402799776509, −1.84657344729815276158260845170, 0.41088917350351442397668150678, 2.77307815064605247674018776904, 4.00182741418541128155360918095, 4.82852093519932152958298672962, 5.24346226219001132319147725938, 6.55952919690188527741239438880, 7.36017519572288887720778203131, 8.337191626585988668390642064764, 8.945674332092985535848650427632, 9.642212634933498465421026902688

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