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Properties

Label 2-980-140.139-c1-0-94
Degree $2$
Conductor $980$
Sign $-0.343 + 0.939i$
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  + 1.41i·2-s − 2.00·4-s + (−0.158 − 2.23i)5-s − 2.82i·8-s + 3·9-s + (3.15 − 0.224i)10-s − 5.99·13-s + 4.00·16-s − 6.62·17-s + 4.24i·18-s + (0.317 + 4.46i)20-s + (−4.94 + 0.707i)25-s − 8.47i·26-s − 9.89·29-s + 5.65i·32-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s + (−0.0708 − 0.997i)5-s − 1.00i·8-s + 9-s + (0.997 − 0.0708i)10-s − 1.66·13-s + 1.00·16-s − 1.60·17-s + 0.999i·18-s + (0.0708 + 0.997i)20-s + (−0.989 + 0.141i)25-s − 1.66i·26-s − 1.83·29-s + 1.00i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.343 + 0.939i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.343 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.190934 - 0.273106i\)
\(L(\frac12)\) \(\approx\) \(0.190934 - 0.273106i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
5 \( 1 + (0.158 + 2.23i)T \)
7 \( 1 \)
good3 \( 1 - 3T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 5.99T + 13T^{2} \)
17 \( 1 + 6.62T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 9.89T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 7.07iT - 37T^{2} \)
41 \( 1 + 3.56iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 14iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 3.11iT - 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−9.517297205484019135802920379109, −8.919262112733134515747376661610, −7.931909517429951765379381641439, −7.25240856761865723176643275987, −6.48153321729390999038324150784, −5.20597830701360282994898312788, −4.71781622134257662350376452112, −3.86169173375896259579069010958, −1.91863374377822642920842364650, −0.14639041080762727431872217361, 1.93866635695927036827263840947, 2.70597825221033768923506814197, 3.97363962383103706354742740207, 4.64201159662751524417689683825, 5.86925510671880584393435267374, 7.17176065036849673018925521979, 7.54179259950133573970286747738, 8.963953257463201638917434552991, 9.622723908561502602009734541388, 10.34151914262841524371682797757

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