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Properties

Label 2-980-140.107-c1-0-29
Degree $2$
Conductor $980$
Sign $0.242 - 0.970i$
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  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (0.448 − 2.19i)5-s + (−1.99 + 2i)8-s + (2.59 + 1.5i)9-s + (0.189 + 3.15i)10-s + (−4.24 + 4.24i)13-s + (1.99 − 3.46i)16-s + (−2.07 + 7.72i)17-s + (−4.09 − 1.09i)18-s + (−1.41 − 4.24i)20-s + (−4.59 − 1.96i)25-s + (4.24 − 7.34i)26-s + 10i·29-s + (−1.46 + 5.46i)32-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s + (0.200 − 0.979i)5-s + (−0.707 + 0.707i)8-s + (0.866 + 0.5i)9-s + (0.0599 + 0.998i)10-s + (−1.17 + 1.17i)13-s + (0.499 − 0.866i)16-s + (−0.502 + 1.87i)17-s + (−0.965 − 0.258i)18-s + (−0.316 − 0.948i)20-s + (−0.919 − 0.392i)25-s + (0.832 − 1.44i)26-s + 1.85i·29-s + (−0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.242 - 0.970i$
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: \((2,\ 980,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685725 + 0.535662i\)
\(L(\frac12)\) \(\approx\) \(0.685725 + 0.535662i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
5 \( 1 + (-0.448 + 2.19i)T \)
7 \( 1 \)
good3 \( 1 + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.24 - 4.24i)T - 13iT^{2} \)
17 \( 1 + (2.07 - 7.72i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 10iT - 29T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.56 + 2.56i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.83 - 1.83i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.707 + 1.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (5.79 + 1.55i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (15.9 + 9.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.65 - 5.65i)T + 97iT^{2} \)
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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

−10.03415468062874006365739892318, −9.244207513290175257731452594563, −8.649687773834674273837570664578, −7.74669449816272879158527221160, −6.99058359708219534748444331962, −6.06354397899796474861247122584, −4.99458929528763931879808475809, −4.14140416900770406868536796839, −2.20426400748469427014285714911, −1.41206417905714831312587793825, 0.56260101775574138935949874188, 2.36897383851846793429427220391, 2.98426470868866535797150868495, 4.33874239712566367194366143754, 5.76235339671260267919870256482, 6.79288942844360145686747291505, 7.35847937418472965821282301605, 7.998928093885487407259157934649, 9.399269162046978845802649382753, 9.790805897154721501857807148550

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