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Properties

Label 2-980-20.3-c1-0-34
Degree $2$
Conductor $980$
Sign $0.850 - 0.525i$
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 − i)2-s − 2i·4-s + (−1 + 2i)5-s + (−2 − 2i)8-s + 3i·9-s + (1 + 3i)10-s + (5 + 5i)13-s − 4·16-s + (−5 + 5i)17-s + (3 + 3i)18-s + (4 + 2i)20-s + (−3 − 4i)25-s + 10·26-s + 4i·29-s + (−4 + 4i)32-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·4-s + (−0.447 + 0.894i)5-s + (−0.707 − 0.707i)8-s + i·9-s + (0.316 + 0.948i)10-s + (1.38 + 1.38i)13-s − 16-s + (−1.21 + 1.21i)17-s + (0.707 + 0.707i)18-s + (0.894 + 0.447i)20-s + (−0.600 − 0.800i)25-s + 1.96·26-s + 0.742i·29-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79332 + 0.509445i\)
\(L(\frac12)\) \(\approx\) \(1.79332 + 0.509445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + (1 - 2i)T \)
7 \( 1 \)
good3 \( 1 - 3iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-5 - 5i)T + 13iT^{2} \)
17 \( 1 + (5 - 5i)T - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-7 + 7i)T - 37iT^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-9 - 9i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + (-5 + 5i)T - 97iT^{2} \)
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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

−10.67599034945331494902288757434, −9.330205966327137550518449663569, −8.562429496725344523077579243404, −7.38671161124681677231404147115, −6.45133865268278004635174474375, −5.83895897040164102096025854604, −4.33533192766915908418348351732, −3.98861786409896714761697870325, −2.65249025480493873713176071184, −1.70548983414259502956387981068, 0.69954960253167344242305936241, 2.86041093076571684453023707476, 3.85994512540815833098706395062, 4.62836210447680006054129101883, 5.67053290322739446642823771052, 6.33769607777875777369798015571, 7.38248269074306429111710550001, 8.241725848428945553568723512438, 8.828725030890345368320182968308, 9.627981487260499818513518465931

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