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Properties

Label 2-980-140.123-c1-0-38
Degree $2$
Conductor $980$
Sign $-0.839 - 0.542i$
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  + (−0.112 + 1.40i)2-s + (0.643 + 2.40i)3-s + (−1.97 − 0.316i)4-s + (1.36 − 1.77i)5-s + (−3.45 + 0.637i)6-s + (0.667 − 2.74i)8-s + (−2.75 + 1.58i)9-s + (2.34 + 2.11i)10-s + (4.62 + 2.67i)11-s + (−0.511 − 4.94i)12-s + (2.61 + 2.61i)13-s + (5.13 + 2.12i)15-s + (3.80 + 1.24i)16-s + (0.381 + 1.42i)17-s + (−1.93 − 4.05i)18-s + (−0.130 − 0.225i)19-s + ⋯
L(s)  = 1  + (−0.0793 + 0.996i)2-s + (0.371 + 1.38i)3-s + (−0.987 − 0.158i)4-s + (0.609 − 0.793i)5-s + (−1.41 + 0.260i)6-s + (0.235 − 0.971i)8-s + (−0.916 + 0.529i)9-s + (0.742 + 0.670i)10-s + (1.39 + 0.805i)11-s + (−0.147 − 1.42i)12-s + (0.726 + 0.726i)13-s + (1.32 + 0.549i)15-s + (0.950 + 0.312i)16-s + (0.0925 + 0.345i)17-s + (−0.454 − 0.956i)18-s + (−0.0298 − 0.0517i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.839 - 0.542i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.839 - 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.533256 + 1.80788i\)
\(L(\frac12)\) \(\approx\) \(0.533256 + 1.80788i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.112 - 1.40i)T \)
5 \( 1 + (-1.36 + 1.77i)T \)
7 \( 1 \)
good3 \( 1 + (-0.643 - 2.40i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-4.62 - 2.67i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.61 - 2.61i)T + 13iT^{2} \)
17 \( 1 + (-0.381 - 1.42i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.130 + 0.225i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.44 + 0.388i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.36iT - 29T^{2} \)
31 \( 1 + (-1.77 - 1.02i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (11.2 + 3.00i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 9.70T + 41T^{2} \)
43 \( 1 + (6.86 - 6.86i)T - 43iT^{2} \)
47 \( 1 + (0.357 - 1.33i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.48 - 0.665i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.26 - 3.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.06 + 3.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.78 + 1.81i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.29iT - 71T^{2} \)
73 \( 1 + (-13.1 + 3.53i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.09 + 8.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.83 - 5.83i)T - 83iT^{2} \)
89 \( 1 + (-6.60 + 3.81i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.5 - 11.5i)T - 97iT^{2} \)
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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

−9.881796168216091022094272162850, −9.316864398361556406554570336536, −8.940738178402837709268855889268, −8.114091675015054938206015904247, −6.77309406649165675752311998475, −6.05713873389013690986497109692, −4.96643540781153130412002620748, −4.32540897475125562738964969192, −3.66784826305040597015906542201, −1.55278769567736899012074188406, 1.00694374477868920650833555746, 1.91855953927789131989167484830, 3.00903888711270579076838521604, 3.75537214744607025279121090587, 5.48293754663752095821685026769, 6.34796460501493171674895619952, 7.11236972577358221629443630185, 8.195113314914620377205606245626, 8.780805708813938853452994757407, 9.710349379274364745729164235619

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