Properties

Label 2-980-140.107-c1-0-31
Degree $2$
Conductor $980$
Sign $-0.905 - 0.425i$
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  + (1.04 + 0.949i)2-s + (−0.0504 + 0.188i)3-s + (0.198 + 1.99i)4-s + (−2.13 − 0.657i)5-s + (−0.231 + 0.149i)6-s + (−1.68 + 2.27i)8-s + (2.56 + 1.48i)9-s + (−1.61 − 2.71i)10-s + (2.70 − 1.56i)11-s + (−0.384 − 0.0631i)12-s + (−2.50 + 2.50i)13-s + (0.231 − 0.369i)15-s + (−3.92 + 0.788i)16-s + (−1.04 + 3.90i)17-s + (1.28 + 3.98i)18-s + (−1.64 + 2.85i)19-s + ⋯
L(s)  = 1  + (0.741 + 0.671i)2-s + (−0.0291 + 0.108i)3-s + (0.0990 + 0.995i)4-s + (−0.955 − 0.294i)5-s + (−0.0945 + 0.0610i)6-s + (−0.594 + 0.804i)8-s + (0.855 + 0.493i)9-s + (−0.511 − 0.859i)10-s + (0.816 − 0.471i)11-s + (−0.111 − 0.0182i)12-s + (−0.694 + 0.694i)13-s + (0.0598 − 0.0953i)15-s + (−0.980 + 0.197i)16-s + (−0.253 + 0.947i)17-s + (0.302 + 0.939i)18-s + (−0.377 + 0.653i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.905 - 0.425i$
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: \((2,\ 980,\ (\ :1/2),\ -0.905 - 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.348051 + 1.55859i\)
\(L(\frac12)\) \(\approx\) \(0.348051 + 1.55859i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 - 0.949i)T \)
5 \( 1 + (2.13 + 0.657i)T \)
7 \( 1 \)
good3 \( 1 + (0.0504 - 0.188i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-2.70 + 1.56i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.50 - 2.50i)T - 13iT^{2} \)
17 \( 1 + (1.04 - 3.90i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.64 - 2.85i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (8.79 - 2.35i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.55iT - 29T^{2} \)
31 \( 1 + (-7.50 + 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.97 - 1.33i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 1.66T + 41T^{2} \)
43 \( 1 + (1.17 + 1.17i)T + 43iT^{2} \)
47 \( 1 + (-0.218 - 0.817i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.53 - 0.411i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.20 + 9.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.40 - 5.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.33 - 2.50i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.15iT - 71T^{2} \)
73 \( 1 + (-8.20 - 2.19i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.0767 + 0.132i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.57 - 8.57i)T + 83iT^{2} \)
89 \( 1 + (-15.0 - 8.70i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.28 + 2.28i)T + 97iT^{2} \)
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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

−10.42147049680368813517821095249, −9.369450487049111935083943991302, −8.305676447236949012653388415198, −7.88133428552852358233546559849, −6.88921629496101289330604426504, −6.17852209478583978748489468777, −4.96961535735792072016549953930, −4.13419576759975459831516732860, −3.67568944767264617726404523460, −1.93160697578922178559046777987, 0.57301518333029801746347796701, 2.18952442731059498584882565516, 3.34364586538992645235469703892, 4.28336583993403096532410884540, 4.83873247954729607278850677314, 6.33123589827008374549671831086, 6.89782551582704696574732173982, 7.80969560456039738829758145291, 9.022695208415656978013214239022, 9.951844380947987316462947930186

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