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Properties

Label 2-980-140.107-c1-0-106
Degree $2$
Conductor $980$
Sign $-0.877 + 0.479i$
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  + (1.19 − 0.763i)2-s + (0.835 − 3.11i)3-s + (0.834 − 1.81i)4-s + (1.91 + 1.15i)5-s + (−1.38 − 4.34i)6-s + (−0.393 − 2.80i)8-s + (−6.41 − 3.70i)9-s + (3.16 − 0.0861i)10-s + (1.21 − 0.703i)11-s + (−4.96 − 4.11i)12-s + (−0.699 + 0.699i)13-s + (5.19 − 5.00i)15-s + (−2.60 − 3.03i)16-s + (−1.19 + 4.46i)17-s + (−10.4 + 0.487i)18-s + (−1.40 + 2.43i)19-s + ⋯
L(s)  = 1  + (0.841 − 0.539i)2-s + (0.482 − 1.79i)3-s + (0.417 − 0.908i)4-s + (0.856 + 0.516i)5-s + (−0.565 − 1.77i)6-s + (−0.139 − 0.990i)8-s + (−2.13 − 1.23i)9-s + (0.999 − 0.0272i)10-s + (0.367 − 0.212i)11-s + (−1.43 − 1.18i)12-s + (−0.193 + 0.193i)13-s + (1.34 − 1.29i)15-s + (−0.651 − 0.758i)16-s + (−0.290 + 1.08i)17-s + (−2.46 + 0.114i)18-s + (−0.321 + 0.557i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 + 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.877 + 0.479i$
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: \((2,\ 980,\ (\ :1/2),\ -0.877 + 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819930 - 3.20739i\)
\(L(\frac12)\) \(\approx\) \(0.819930 - 3.20739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.763i)T \)
5 \( 1 + (-1.91 - 1.15i)T \)
7 \( 1 \)
good3 \( 1 + (-0.835 + 3.11i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.21 + 0.703i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.699 - 0.699i)T - 13iT^{2} \)
17 \( 1 + (1.19 - 4.46i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.40 - 2.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.03 + 0.543i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.85iT - 29T^{2} \)
31 \( 1 + (-3.18 + 1.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.99 + 2.14i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.53T + 41T^{2} \)
43 \( 1 + (2.49 + 2.49i)T + 43iT^{2} \)
47 \( 1 + (-0.684 - 2.55i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.951 - 0.254i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.53 - 6.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.09 - 1.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.08 + 0.827i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + (6.40 + 1.71i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.299 + 0.518i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.53 - 4.53i)T + 83iT^{2} \)
89 \( 1 + (-0.401 - 0.231i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.00 - 8.00i)T + 97iT^{2} \)
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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

−9.704015809909606881176885563469, −8.797080013429206116637076208732, −7.76512094617238945985055024003, −6.83271921927404963901689378271, −6.21487885392991052695630563940, −5.71680383748386533723733699804, −4.04459957767031227568779659943, −2.80654053420388893250561474536, −2.13823004977465466133761757414, −1.18298732109696096641494666270, 2.46214183139906435981272308958, 3.27894115178699843421293213857, 4.53835335374421488770981577943, 4.83509653810647847090916808236, 5.69666709303820689496717262130, 6.73676607931873758490168991165, 8.004583994917630564986209364451, 8.951274056878361063486022371214, 9.323249363599442963647616017677, 10.23952061966761110037720106850

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