Properties

Label 2-980-140.67-c1-0-24
Degree $2$
Conductor $980$
Sign $-0.237 - 0.971i$
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  + (−0.366 + 1.36i)2-s + (−1.73 − i)4-s + (−1.86 − 1.23i)5-s + (2 − 1.99i)8-s + (−2.59 + 1.5i)9-s + (2.36 − 2.09i)10-s + (1 − i)13-s + (1.99 + 3.46i)16-s + (4.09 − 1.09i)17-s + (−1.09 − 4.09i)18-s + (2 + 4i)20-s + (1.96 + 4.59i)25-s + (1 + 1.73i)26-s + 4i·29-s + (−5.46 + 1.46i)32-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)4-s + (−0.834 − 0.550i)5-s + (0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.748 − 0.663i)10-s + (0.277 − 0.277i)13-s + (0.499 + 0.866i)16-s + (0.993 − 0.266i)17-s + (−0.258 − 0.965i)18-s + (0.447 + 0.894i)20-s + (0.392 + 0.919i)25-s + (0.196 + 0.339i)26-s + 0.742i·29-s + (−0.965 + 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.237 - 0.971i$
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: \((2,\ 980,\ (\ :1/2),\ -0.237 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.517954 + 0.660073i\)
\(L(\frac12)\) \(\approx\) \(0.517954 + 0.660073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
5 \( 1 + (1.86 + 1.23i)T \)
7 \( 1 \)
good3 \( 1 + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (-4.09 + 1.09i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.56 - 9.56i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.29 - 12.2i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-4.02 - 15.0i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-13.8 + 8i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13 + 13i)T + 97iT^{2} \)
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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

−10.09768436472000509853174854508, −9.071034376482838217491851202417, −8.435666294431672199598254200965, −7.81663762264873537965298712483, −7.06803307519818886681158591176, −5.84560053950339525983148681579, −5.21702369564741684539490114235, −4.27463355152783802809991195544, −3.11798097105132635371657819136, −1.04080545308933562506111898808, 0.54888306258959967741179582848, 2.30196820499498107034285342555, 3.41026813961926222205522704825, 3.95104313842934157132971414662, 5.25732030553520661090493643061, 6.35294445817611710197647100692, 7.55452933186477464361343644250, 8.190034773308068240033329344737, 9.026774763357027835535211586563, 9.833059421083350282518883607420

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