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Properties

Label 2-980-140.67-c1-0-29
Degree $2$
Conductor $980$
Sign $-0.593 - 0.805i$
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  + (0.805 + 1.16i)2-s + (−3.15 + 0.844i)3-s + (−0.703 + 1.87i)4-s + (−0.260 − 2.22i)5-s + (−3.52 − 2.98i)6-s + (−2.74 + 0.689i)8-s + (6.62 − 3.82i)9-s + (2.37 − 2.09i)10-s + (1.96 + 1.13i)11-s + (0.635 − 6.49i)12-s + (1.38 − 1.38i)13-s + (2.69 + 6.78i)15-s + (−3.01 − 2.63i)16-s + (−0.186 + 0.0499i)17-s + (9.78 + 4.62i)18-s + (3.45 + 5.98i)19-s + ⋯
L(s)  = 1  + (0.569 + 0.822i)2-s + (−1.81 + 0.487i)3-s + (−0.351 + 0.936i)4-s + (−0.116 − 0.993i)5-s + (−1.43 − 1.21i)6-s + (−0.969 + 0.243i)8-s + (2.20 − 1.27i)9-s + (0.750 − 0.661i)10-s + (0.593 + 0.342i)11-s + (0.183 − 1.87i)12-s + (0.383 − 0.383i)13-s + (0.696 + 1.75i)15-s + (−0.752 − 0.658i)16-s + (−0.0451 + 0.0121i)17-s + (2.30 + 1.08i)18-s + (0.792 + 1.37i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.593 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.593 - 0.805i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.593 - 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.434269 + 0.859375i\)
\(L(\frac12)\) \(\approx\) \(0.434269 + 0.859375i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.805 - 1.16i)T \)
5 \( 1 + (0.260 + 2.22i)T \)
7 \( 1 \)
good3 \( 1 + (3.15 - 0.844i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.96 - 1.13i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.38 + 1.38i)T - 13iT^{2} \)
17 \( 1 + (0.186 - 0.0499i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.45 - 5.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.866 + 3.23i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.33iT - 29T^{2} \)
31 \( 1 + (0.430 + 0.248i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.904 + 3.37i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.22T + 41T^{2} \)
43 \( 1 + (2.91 + 2.91i)T + 43iT^{2} \)
47 \( 1 + (2.40 + 0.645i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.87 - 6.98i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.61 - 4.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.00 - 8.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.21 - 12.0i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.60iT - 71T^{2} \)
73 \( 1 + (-3.38 - 12.6i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.66 - 9.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.591 + 0.591i)T + 83iT^{2} \)
89 \( 1 + (-3.45 + 1.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.09 + 1.09i)T + 97iT^{2} \)
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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

−10.32922785023187349981177882764, −9.472679660891283304745173179995, −8.598029420348758065872538991256, −7.47010874799429363914405338068, −6.63808285461829269090794026788, −5.69173833362550135702731362216, −5.34418952417490404518538677654, −4.38000139879729101727031468706, −3.74835381928925044828831688422, −1.09404189236223748914786342805, 0.59182551506647517843409121115, 1.91983314126946747215255612995, 3.37951685175502599305870706881, 4.50547089033427457514743061719, 5.35008927866693357068054999790, 6.35342114077628943887658421509, 6.60160024390806552192115268658, 7.69897678285084717910774328245, 9.366219766907498954342111316328, 10.06991602622884063160034071122

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