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Properties

Label 2-980-140.107-c1-0-90
Degree $2$
Conductor $980$
Sign $0.900 + 0.434i$
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  + (1.24 − 0.667i)2-s + (−0.543 + 2.02i)3-s + (1.10 − 1.66i)4-s + (1.62 − 1.53i)5-s + (0.676 + 2.89i)6-s + (0.270 − 2.81i)8-s + (−1.22 − 0.704i)9-s + (0.998 − 3.00i)10-s + (−0.366 + 0.211i)11-s + (2.77 + 3.15i)12-s + (1.56 − 1.56i)13-s + (2.23 + 4.12i)15-s + (−1.54 − 3.69i)16-s + (−1.18 + 4.41i)17-s + (−1.99 − 0.0635i)18-s + (3.66 − 6.35i)19-s + ⋯
L(s)  = 1  + (0.881 − 0.472i)2-s + (−0.313 + 1.17i)3-s + (0.554 − 0.832i)4-s + (0.726 − 0.687i)5-s + (0.276 + 1.18i)6-s + (0.0955 − 0.995i)8-s + (−0.406 − 0.234i)9-s + (0.315 − 0.948i)10-s + (−0.110 + 0.0637i)11-s + (0.800 + 0.910i)12-s + (0.433 − 0.433i)13-s + (0.576 + 1.06i)15-s + (−0.385 − 0.922i)16-s + (−0.286 + 1.07i)17-s + (−0.469 − 0.0149i)18-s + (0.841 − 1.45i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.900 + 0.434i$
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: \((2,\ 980,\ (\ :1/2),\ 0.900 + 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.79292 - 0.639236i\)
\(L(\frac12)\) \(\approx\) \(2.79292 - 0.639236i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 + 0.667i)T \)
5 \( 1 + (-1.62 + 1.53i)T \)
7 \( 1 \)
good3 \( 1 + (0.543 - 2.02i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.366 - 0.211i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.56 + 1.56i)T - 13iT^{2} \)
17 \( 1 + (1.18 - 4.41i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.66 + 6.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.44 + 1.72i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.72iT - 29T^{2} \)
31 \( 1 + (-1.70 + 0.982i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.20 - 1.92i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.01T + 41T^{2} \)
43 \( 1 + (1.88 + 1.88i)T + 43iT^{2} \)
47 \( 1 + (-1.70 - 6.35i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.780 - 0.209i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.35 - 2.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.925 - 1.60i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.70 - 2.33i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.16iT - 71T^{2} \)
73 \( 1 + (6.67 + 1.78i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.77 - 3.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.71 - 4.71i)T + 83iT^{2} \)
89 \( 1 + (10.0 + 5.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.64 - 4.64i)T + 97iT^{2} \)
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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

−10.17029021658513881294610599397, −9.326340834855242462319444659092, −8.705708920583962514705013653013, −7.12275012897246930020519218588, −6.11176861928020810782001430775, −5.17938224895652524401673857746, −4.83817711669599334576843283206, −3.81081591745858395965010156588, −2.72661765705343891614837032406, −1.22554148377945736427466262342, 1.55361877497468337222927388267, 2.64523957682855554453810144566, 3.71195981429725237215629169164, 5.17825484476626766319685699579, 5.87509216370269297053779292636, 6.69683398555617020783681720979, 7.17504435349229275303203418967, 7.945084925498136461225338891880, 9.119220428725583342669956384057, 10.16711906776526739441620932590

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