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Properties

Label 2-980-140.107-c1-0-107
Degree $2$
Conductor $980$
Sign $-0.768 + 0.640i$
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  + (1.41 + 0.0657i)2-s + (0.835 − 3.11i)3-s + (1.99 + 0.185i)4-s + (−1.91 − 1.15i)5-s + (1.38 − 4.34i)6-s + (2.80 + 0.393i)8-s + (−6.41 − 3.70i)9-s + (−2.62 − 1.75i)10-s + (−1.21 + 0.703i)11-s + (2.24 − 6.05i)12-s + (0.699 − 0.699i)13-s + (−5.19 + 5.00i)15-s + (3.93 + 0.739i)16-s + (1.19 − 4.46i)17-s + (−8.82 − 5.65i)18-s + (−1.40 + 2.43i)19-s + ⋯
L(s)  = 1  + (0.998 + 0.0464i)2-s + (0.482 − 1.79i)3-s + (0.995 + 0.0928i)4-s + (−0.856 − 0.516i)5-s + (0.565 − 1.77i)6-s + (0.990 + 0.139i)8-s + (−2.13 − 1.23i)9-s + (−0.831 − 0.555i)10-s + (−0.367 + 0.212i)11-s + (0.647 − 1.74i)12-s + (0.193 − 0.193i)13-s + (−1.34 + 1.29i)15-s + (0.982 + 0.184i)16-s + (0.290 − 1.08i)17-s + (−2.07 − 1.33i)18-s + (−0.321 + 0.557i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.768 + 0.640i$
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: \((2,\ 980,\ (\ :1/2),\ -0.768 + 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.955252 - 2.63817i\)
\(L(\frac12)\) \(\approx\) \(0.955252 - 2.63817i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0657i)T \)
5 \( 1 + (1.91 + 1.15i)T \)
7 \( 1 \)
good3 \( 1 + (-0.835 + 3.11i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.21 - 0.703i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.699 + 0.699i)T - 13iT^{2} \)
17 \( 1 + (-1.19 + 4.46i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.40 - 2.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.03 - 0.543i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.85iT - 29T^{2} \)
31 \( 1 + (-3.18 + 1.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.99 + 2.14i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 3.53T + 41T^{2} \)
43 \( 1 + (-2.49 - 2.49i)T + 43iT^{2} \)
47 \( 1 + (-0.684 - 2.55i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.951 - 0.254i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.53 - 6.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.09 + 1.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.08 - 0.827i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + (-6.40 - 1.71i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.299 - 0.518i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.53 - 4.53i)T + 83iT^{2} \)
89 \( 1 + (0.401 + 0.231i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.00 + 8.00i)T + 97iT^{2} \)
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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

−9.534140271363875630088599923052, −8.244716958155905729952911264398, −7.85744158033221078781019844542, −7.21293611406149345273716015639, −6.31694961734289913694640913634, −5.47905384409080387751834186433, −4.25887658435907510544500187027, −3.12085076111780627625774956385, −2.21164617682986884771644990257, −0.885956940978310726440837332418, 2.53423115378390903623976416370, 3.40587479598073797253053257844, 4.03973186182466638002763016901, 4.76235938695154098967768177463, 5.70656113549655797012213197847, 6.76059003511915026367297196703, 8.013858017571901754305290101470, 8.549420975807471221059640245249, 9.800711631366675609016324274625, 10.53652568491666751671702384427

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