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Properties

Label 2-980-140.107-c1-0-104
Degree $2$
Conductor $980$
Sign $-0.951 + 0.308i$
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  + (0.996 − 1.00i)2-s + (0.463 − 1.72i)3-s + (−0.0138 − 1.99i)4-s + (1.91 − 1.16i)5-s + (−1.27 − 2.18i)6-s + (−2.02 − 1.97i)8-s + (−0.177 − 0.102i)9-s + (0.737 − 3.07i)10-s + (−1.83 + 1.05i)11-s + (−3.46 − 0.902i)12-s + (−2.86 + 2.86i)13-s + (−1.12 − 3.84i)15-s + (−3.99 + 0.0552i)16-s + (1.63 − 6.10i)17-s + (−0.279 + 0.0759i)18-s + (1.47 − 2.55i)19-s + ⋯
L(s)  = 1  + (0.704 − 0.709i)2-s + (0.267 − 0.998i)3-s + (−0.00690 − 0.999i)4-s + (0.854 − 0.519i)5-s + (−0.519 − 0.893i)6-s + (−0.714 − 0.699i)8-s + (−0.0591 − 0.0341i)9-s + (0.233 − 0.972i)10-s + (−0.551 + 0.318i)11-s + (−1.00 − 0.260i)12-s + (−0.794 + 0.794i)13-s + (−0.290 − 0.991i)15-s + (−0.999 + 0.0138i)16-s + (0.396 − 1.48i)17-s + (−0.0659 + 0.0179i)18-s + (0.338 − 0.586i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.951 + 0.308i$
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: \((2,\ 980,\ (\ :1/2),\ -0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.418948 - 2.65227i\)
\(L(\frac12)\) \(\approx\) \(0.418948 - 2.65227i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.996 + 1.00i)T \)
5 \( 1 + (-1.91 + 1.16i)T \)
7 \( 1 \)
good3 \( 1 + (-0.463 + 1.72i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.83 - 1.05i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.86 - 2.86i)T - 13iT^{2} \)
17 \( 1 + (-1.63 + 6.10i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.47 + 2.55i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.38 - 1.44i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.86iT - 29T^{2} \)
31 \( 1 + (-5.20 + 3.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.34 - 1.43i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.98T + 41T^{2} \)
43 \( 1 + (-5.64 - 5.64i)T + 43iT^{2} \)
47 \( 1 + (-1.57 - 5.87i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.766 - 0.205i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.48 - 4.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.843 + 1.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.0 - 2.95i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 0.610iT - 71T^{2} \)
73 \( 1 + (4.41 + 1.18i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.12 + 12.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.51 + 5.51i)T + 83iT^{2} \)
89 \( 1 + (-1.80 - 1.03i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.95 - 1.95i)T + 97iT^{2} \)
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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

−9.673569327772941909154325803251, −9.113497892399466313261520093965, −7.76654733842271119016317334603, −7.01064898442031297673243111128, −6.10356826698537235769211573269, −5.09861101100263923032966255747, −4.45704811659517047341322267891, −2.71964091142537908465183808640, −2.15358308623785926420804981007, −0.972986801219964943165859312758, 2.34835296941429426333960752951, 3.37894910522940498065095588466, 4.16481908489591775710704808754, 5.34814404306138779175939149323, 5.82568689764302476034202073207, 6.84337126014971503664098802753, 7.88277823350036651577520439993, 8.567157241671022394497088045812, 9.725474625097960599278662132313, 10.22774156844986127841538379732

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