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Properties

Label 2-980-35.27-c1-0-3
Degree $2$
Conductor $980$
Sign $0.962 - 0.271i$
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  + (−2.28 − 2.28i)3-s + (−1.22 − 1.87i)5-s + 7.46i·9-s − 1.23·11-s + (2.44 + 2.44i)13-s + (−1.48 + 7.07i)15-s + (−1.08 + 1.08i)17-s − 4.29·19-s + (−0.731 + 0.731i)23-s + (−2.01 + 4.57i)25-s + (10.2 − 10.2i)27-s + 3.02i·29-s + 5.40i·31-s + (2.81 + 2.81i)33-s + (0.896 + 0.896i)37-s + ⋯
L(s)  = 1  + (−1.32 − 1.32i)3-s + (−0.546 − 0.837i)5-s + 2.48i·9-s − 0.370·11-s + (0.677 + 0.677i)13-s + (−0.384 + 1.82i)15-s + (−0.262 + 0.262i)17-s − 0.984·19-s + (−0.152 + 0.152i)23-s + (−0.402 + 0.915i)25-s + (1.96 − 1.96i)27-s + 0.562i·29-s + 0.970i·31-s + (0.489 + 0.489i)33-s + (0.147 + 0.147i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.962 - 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489147 + 0.0676045i\)
\(L(\frac12)\) \(\approx\) \(0.489147 + 0.0676045i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.22 + 1.87i)T \)
7 \( 1 \)
good3 \( 1 + (2.28 + 2.28i)T + 3iT^{2} \)
11 \( 1 + 1.23T + 11T^{2} \)
13 \( 1 + (-2.44 - 2.44i)T + 13iT^{2} \)
17 \( 1 + (1.08 - 1.08i)T - 17iT^{2} \)
19 \( 1 + 4.29T + 19T^{2} \)
23 \( 1 + (0.731 - 0.731i)T - 23iT^{2} \)
29 \( 1 - 3.02iT - 29T^{2} \)
31 \( 1 - 5.40iT - 31T^{2} \)
37 \( 1 + (-0.896 - 0.896i)T + 37iT^{2} \)
41 \( 1 - 1.26iT - 41T^{2} \)
43 \( 1 + (-6.08 + 6.08i)T - 43iT^{2} \)
47 \( 1 + (-3.92 + 3.92i)T - 47iT^{2} \)
53 \( 1 + (-8.79 + 8.79i)T - 53iT^{2} \)
59 \( 1 + 3.43T + 59T^{2} \)
61 \( 1 - 5.56iT - 61T^{2} \)
67 \( 1 + (-2.96 - 2.96i)T + 67iT^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + (-3.71 - 3.71i)T + 73iT^{2} \)
79 \( 1 - 5.02iT - 79T^{2} \)
83 \( 1 + (8.84 + 8.84i)T + 83iT^{2} \)
89 \( 1 - 6.11T + 89T^{2} \)
97 \( 1 + (11.1 - 11.1i)T - 97iT^{2} \)
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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

−10.42150780978015628050383651949, −8.886732097575460662771665021131, −8.290833932416989255411002904236, −7.34072289834093675453385105493, −6.66936008284688123191774823571, −5.78290323266211179815153776820, −5.02261845432686044724676108027, −4.02131592552449180611370854329, −2.08811467389090881565019976882, −1.02350584761957903596950892942, 0.33669423257023511870683137088, 2.83766567665480949828156628219, 3.97911061029912839193706793830, 4.51675219954122590690097843424, 5.80222300229101637239950396611, 6.17975074149995976235412610870, 7.29927543404191426592231533001, 8.373326650832279434104054774395, 9.444622510200758903049897588578, 10.23249188615624369204788689898

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