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Properties

Label 2-980-20.19-c0-0-0
Degree $2$
Conductor $980$
Sign $0.707 - 0.707i$
Analytic cond. $0.489083$
Root an. cond. $0.699345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + (−0.707 + 0.707i)5-s + i·8-s − 9-s + (0.707 + 0.707i)10-s + 1.41i·13-s + 16-s + 1.41i·17-s + i·18-s + (0.707 − 0.707i)20-s − 1.00i·25-s + 1.41·26-s − i·32-s + 1.41·34-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−0.707 + 0.707i)5-s + i·8-s − 9-s + (0.707 + 0.707i)10-s + 1.41i·13-s + 16-s + 1.41i·17-s + i·18-s + (0.707 − 0.707i)20-s − 1.00i·25-s + 1.41·26-s − i·32-s + 1.41·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5568104354\)
\(L(\frac12)\) \(\approx\) \(0.5568104354\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 \)
good3 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 2iT - T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + 1.41iT - T^{2} \)
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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

−10.46238853933234341838056300457, −9.667332529496313533860220420156, −8.544680885048340428591286843487, −8.240522059169689902813379734440, −6.89604458435717238916698407724, −6.01430082940369843112638691450, −4.74815458656768011716886465042, −3.83947385823145028259015544178, −3.02755718847270443124899092917, −1.84120496562006941454486017563, 0.53534013729196280025652651376, 2.98530253044838178819832232157, 4.01903251288157073759077382382, 5.30304970252868633953811365137, 5.45404758597821470960295222444, 6.84961743495114197441092533626, 7.68497850518753325857033312683, 8.321375610689721246121577038221, 8.983558221277207994782848661729, 9.807968636322277987689931640934

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