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Properties

Label 2-980-140.139-c1-0-27
Degree $2$
Conductor $980$
Sign $-0.284 - 0.958i$
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  + (−1.29 + 0.561i)2-s + 1.88i·3-s + (1.37 − 1.45i)4-s + (−2.11 + 0.724i)5-s + (−1.05 − 2.44i)6-s + (−0.960 + 2.66i)8-s − 0.534·9-s + (2.33 − 2.12i)10-s − 1.38i·11-s + (2.73 + 2.57i)12-s + 5.79·13-s + (−1.36 − 3.97i)15-s + (−0.245 − 3.99i)16-s + 4.66·17-s + (0.694 − 0.300i)18-s + 5.49·19-s + ⋯
L(s)  = 1  + (−0.917 + 0.396i)2-s + 1.08i·3-s + (0.685 − 0.728i)4-s + (−0.946 + 0.323i)5-s + (−0.430 − 0.996i)6-s + (−0.339 + 0.940i)8-s − 0.178·9-s + (0.739 − 0.672i)10-s − 0.417i·11-s + (0.790 + 0.743i)12-s + 1.60·13-s + (−0.351 − 1.02i)15-s + (−0.0614 − 0.998i)16-s + 1.13·17-s + (0.163 − 0.0707i)18-s + 1.26·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.284 - 0.958i$
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: \((2,\ 980,\ (\ :1/2),\ -0.284 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.587516 + 0.787506i\)
\(L(\frac12)\) \(\approx\) \(0.587516 + 0.787506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 - 0.561i)T \)
5 \( 1 + (2.11 - 0.724i)T \)
7 \( 1 \)
good3 \( 1 - 1.88iT - 3T^{2} \)
11 \( 1 + 1.38iT - 11T^{2} \)
13 \( 1 - 5.79T + 13T^{2} \)
17 \( 1 - 4.66T + 17T^{2} \)
19 \( 1 - 5.49T + 19T^{2} \)
23 \( 1 + 4.64T + 23T^{2} \)
29 \( 1 - 2.04T + 29T^{2} \)
31 \( 1 + 1.66T + 31T^{2} \)
37 \( 1 + 0.285iT - 37T^{2} \)
41 \( 1 - 8.21iT - 41T^{2} \)
43 \( 1 + 3.30T + 43T^{2} \)
47 \( 1 - 0.138iT - 47T^{2} \)
53 \( 1 + 6.72iT - 53T^{2} \)
59 \( 1 - 1.66T + 59T^{2} \)
61 \( 1 - 2.88iT - 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 2.29iT - 71T^{2} \)
73 \( 1 + 4.66T + 73T^{2} \)
79 \( 1 - 10.2iT - 79T^{2} \)
83 \( 1 - 0.0814iT - 83T^{2} \)
89 \( 1 - 3.71iT - 89T^{2} \)
97 \( 1 + 3.88T + 97T^{2} \)
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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

−10.08181150475493928047629427602, −9.548585047975931244342315671921, −8.466053761696553823111693799845, −8.021314723857435417320593717773, −7.03164420600903369217646706932, −6.02769181305417298946168669954, −5.13107205457030224873114124158, −3.85169062905275587706208556476, −3.17090610772707093583259652470, −1.13062618457612939798569514967, 0.817463139987499054512528452912, 1.67396220303921963961190617379, 3.21018149639764768317380376259, 4.04933146357121339838241440376, 5.69380793729555250143267388289, 6.74648524720514993124044205822, 7.50612929333655895370124290345, 8.014770658953614670702174391429, 8.715097023299459102054607594880, 9.718799664977821594590096642693

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