Properties

Label 2-980-140.107-c1-0-26
Degree $2$
Conductor $980$
Sign $-0.844 - 0.535i$
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 − 0.674i)2-s + (−0.664 + 2.47i)3-s + (1.09 + 1.67i)4-s + (1.93 + 1.11i)5-s + (2.49 − 2.63i)6-s + (−0.224 − 2.81i)8-s + (−3.10 − 1.79i)9-s + (−1.65 − 2.69i)10-s + (−1.59 + 0.921i)11-s + (−4.88 + 1.58i)12-s + (2.94 − 2.94i)13-s + (−4.04 + 4.06i)15-s + (−1.62 + 3.65i)16-s + (−0.795 + 2.96i)17-s + (2.65 + 4.32i)18-s + (−2.66 + 4.61i)19-s + ⋯
L(s)  = 1  + (−0.878 − 0.476i)2-s + (−0.383 + 1.43i)3-s + (0.545 + 0.838i)4-s + (0.867 + 0.497i)5-s + (1.01 − 1.07i)6-s + (−0.0793 − 0.996i)8-s + (−1.03 − 0.597i)9-s + (−0.524 − 0.851i)10-s + (−0.481 + 0.277i)11-s + (−1.40 + 0.458i)12-s + (0.817 − 0.817i)13-s + (−1.04 + 1.05i)15-s + (−0.405 + 0.914i)16-s + (−0.192 + 0.720i)17-s + (0.625 + 1.01i)18-s + (−0.611 + 1.05i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.844 - 0.535i$
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: \((2,\ 980,\ (\ :1/2),\ -0.844 - 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.230607 + 0.794703i\)
\(L(\frac12)\) \(\approx\) \(0.230607 + 0.794703i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.674i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 \)
good3 \( 1 + (0.664 - 2.47i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.59 - 0.921i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.94 + 2.94i)T - 13iT^{2} \)
17 \( 1 + (0.795 - 2.96i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.66 - 4.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.44 + 0.654i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.35iT - 29T^{2} \)
31 \( 1 + (3.78 - 2.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.86 + 1.03i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 + (1.02 + 1.02i)T + 43iT^{2} \)
47 \( 1 + (0.210 + 0.785i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.61 + 0.699i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.11 - 3.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.00 + 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.19 + 0.856i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (1.97 + 0.529i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.95 - 6.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.227 - 0.227i)T + 83iT^{2} \)
89 \( 1 + (-3.75 - 2.16i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.196 + 0.196i)T + 97iT^{2} \)
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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

−10.38746657644624046235421427197, −9.817339556799845897393212114441, −8.901812939683384184461587538435, −8.235490751772837910391551652060, −6.96861838309991239488921633417, −6.00080662161796523593132917837, −5.15742103088773432243290264230, −3.83778091312747424167468797724, −3.11315598244682865797673128992, −1.71664405769892306392442687560, 0.53089807369242464448870473404, 1.65311292666505690732592885656, 2.51450129832012605216902325928, 4.79277199772077011485556741608, 5.77883584198855956053146429269, 6.43287906304286799642534992899, 7.02669558619866498761849860337, 7.962251748707826389565043494622, 8.770376129722406513777615271882, 9.379592463018758704295132279284

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