Properties

Label 2-980-140.107-c1-0-25
Degree $2$
Conductor $980$
Sign $-0.264 + 0.964i$
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  + (0.607 + 1.27i)2-s + (−0.643 + 2.40i)3-s + (−1.26 + 1.55i)4-s + (1.36 + 1.77i)5-s + (−3.45 + 0.637i)6-s + (−2.74 − 0.667i)8-s + (−2.75 − 1.58i)9-s + (−1.43 + 2.81i)10-s + (−4.62 + 2.67i)11-s + (−2.91 − 4.02i)12-s + (2.61 − 2.61i)13-s + (−5.13 + 2.12i)15-s + (−0.818 − 3.91i)16-s + (0.381 − 1.42i)17-s + (0.356 − 4.47i)18-s + (0.130 − 0.225i)19-s + ⋯
L(s)  = 1  + (0.429 + 0.902i)2-s + (−0.371 + 1.38i)3-s + (−0.630 + 0.776i)4-s + (0.609 + 0.793i)5-s + (−1.41 + 0.260i)6-s + (−0.971 − 0.235i)8-s + (−0.916 − 0.529i)9-s + (−0.454 + 0.890i)10-s + (−1.39 + 0.805i)11-s + (−0.841 − 1.16i)12-s + (0.726 − 0.726i)13-s + (−1.32 + 0.549i)15-s + (−0.204 − 0.978i)16-s + (0.0925 − 0.345i)17-s + (0.0839 − 1.05i)18-s + (0.0298 − 0.0517i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.264 + 0.964i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.264 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.767621 - 1.00630i\)
\(L(\frac12)\) \(\approx\) \(0.767621 - 1.00630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.607 - 1.27i)T \)
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

−10.59754976906156186479218473206, −9.746173486577382313077138387737, −9.091336407317014598194954059620, −7.924137227907711638719762822628, −7.15513336588109732701546082359, −6.05603509846885550276168689296, −5.35100375015222714279493586323, −4.74735988501915504290256439417, −3.59349847657235956068428915247, −2.73423424374807742183431036195, 0.52695079545592687287013267415, 1.63153117827448850146556260273, 2.49740771805808078950306491765, 3.92325651131388121648768183567, 5.28111389050161521918461752212, 5.74852092380241268477099072588, 6.59827899049491453047556047651, 7.84761455714914443193515275778, 8.637106024261341139938703448492, 9.405675362763626824012778898029

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