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Properties

Label 2-980-980.319-c0-0-0
Degree $2$
Conductor $980$
Sign $-0.918 + 0.394i$
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  + (−0.955 + 0.294i)2-s + (0.722 + 1.84i)3-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (−1.23 − 1.54i)6-s + (−0.826 + 0.563i)7-s + (−0.623 + 0.781i)8-s + (−2.13 + 1.98i)9-s + (0.988 − 0.149i)10-s + (1.63 + 1.11i)12-s + (0.623 − 0.781i)14-s + (−0.440 − 1.92i)15-s + (0.365 − 0.930i)16-s + (1.45 − 2.52i)18-s + (−0.900 + 0.433i)20-s + (−1.63 − 1.11i)21-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)2-s + (0.722 + 1.84i)3-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (−1.23 − 1.54i)6-s + (−0.826 + 0.563i)7-s + (−0.623 + 0.781i)8-s + (−2.13 + 1.98i)9-s + (0.988 − 0.149i)10-s + (1.63 + 1.11i)12-s + (0.623 − 0.781i)14-s + (−0.440 − 1.92i)15-s + (0.365 − 0.930i)16-s + (1.45 − 2.52i)18-s + (−0.900 + 0.433i)20-s + (−1.63 − 1.11i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.918 + 0.394i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ -0.918 + 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4499753782\)
\(L(\frac12)\) \(\approx\) \(0.4499753782\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.955 - 0.294i)T \)
5 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (0.826 - 0.563i)T \)
good3 \( 1 + (-0.722 - 1.84i)T + (-0.733 + 0.680i)T^{2} \)
11 \( 1 + (-0.0747 - 0.997i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.988 + 0.149i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.0546 + 0.728i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (0.134 - 0.0648i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.365 - 0.930i)T^{2} \)
41 \( 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.914 - 1.14i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (1.19 - 0.367i)T + (0.826 - 0.563i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (-0.603 - 0.411i)T + (0.365 + 0.930i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.826 - 0.563i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.367 - 1.61i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.109 - 0.101i)T + (0.0747 - 0.997i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.34387717145774975307329078724, −9.710913642383523724103140934455, −9.049811747809149786663088107672, −8.432629440686482003633925609463, −7.80188228818486425086702320923, −6.49518590082587334887980870803, −5.39361271718417493629636512745, −4.46681263097504797843796060117, −3.36804999460949874094967306368, −2.62587891490834548379247779486, 0.50892856941881825044948085817, 1.88895369418691111827214283198, 3.09820862807257548867945807563, 3.67374650949343601936854426869, 6.00153485003995407103589945641, 6.95168731855343069451687176068, 7.24515935287108056728233038512, 8.003626377202204278804825950318, 8.691161839296284018777417185237, 9.456107922668132715819091990447

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