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Properties

Label 2-980-28.27-c1-0-37
Degree $2$
Conductor $980$
Sign $0.972 + 0.234i$
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  + (−0.431 − 1.34i)2-s + 2.73·3-s + (−1.62 + 1.16i)4-s + i·5-s + (−1.18 − 3.68i)6-s + (2.26 + 1.69i)8-s + 4.49·9-s + (1.34 − 0.431i)10-s − 0.100i·11-s + (−4.45 + 3.18i)12-s + 4.11i·13-s + 2.73i·15-s + (1.29 − 3.78i)16-s + 5.39i·17-s + (−1.93 − 6.05i)18-s + 7.45·19-s + ⋯
L(s)  = 1  + (−0.305 − 0.952i)2-s + 1.58·3-s + (−0.813 + 0.581i)4-s + 0.447i·5-s + (−0.482 − 1.50i)6-s + (0.801 + 0.597i)8-s + 1.49·9-s + (0.425 − 0.136i)10-s − 0.0302i·11-s + (−1.28 + 0.918i)12-s + 1.14i·13-s + 0.706i·15-s + (0.324 − 0.945i)16-s + 1.30i·17-s + (−0.456 − 1.42i)18-s + 1.71·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.972 + 0.234i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.972 + 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18522 - 0.260270i\)
\(L(\frac12)\) \(\approx\) \(2.18522 - 0.260270i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.431 + 1.34i)T \)
5 \( 1 - iT \)
7 \( 1 \)
good3 \( 1 - 2.73T + 3T^{2} \)
11 \( 1 + 0.100iT - 11T^{2} \)
13 \( 1 - 4.11iT - 13T^{2} \)
17 \( 1 - 5.39iT - 17T^{2} \)
19 \( 1 - 7.45T + 19T^{2} \)
23 \( 1 - 1.50iT - 23T^{2} \)
29 \( 1 + 2.37T + 29T^{2} \)
31 \( 1 + 5.44T + 31T^{2} \)
37 \( 1 - 1.03T + 37T^{2} \)
41 \( 1 + 7.99iT - 41T^{2} \)
43 \( 1 + 7.04iT - 43T^{2} \)
47 \( 1 - 4.44T + 47T^{2} \)
53 \( 1 - 6.14T + 53T^{2} \)
59 \( 1 - 8.52T + 59T^{2} \)
61 \( 1 + 7.90iT - 61T^{2} \)
67 \( 1 - 0.109iT - 67T^{2} \)
71 \( 1 - 6.73iT - 71T^{2} \)
73 \( 1 + 6.14iT - 73T^{2} \)
79 \( 1 - 4.27iT - 79T^{2} \)
83 \( 1 - 6.50T + 83T^{2} \)
89 \( 1 - 3.19iT - 89T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−9.791479665363594977715563272830, −9.179868663020377513593104798631, −8.589380426640651138674015531127, −7.67346021476486395317565260089, −7.06878483829745946742259977001, −5.43798910352735752323356525266, −3.92042269509667148517289196158, −3.60522106835359230482331808353, −2.43487672800167087273396131219, −1.63215786295818948934965192801, 1.06392150944329982998444190262, 2.71988592632244488697546733672, 3.70247337991162002946968602361, 4.89632328300167431726127820670, 5.66877906720526979557786213291, 7.09568584331847285589311868809, 7.68392540116267843211724394670, 8.231995326416360431527795167652, 9.220188205367784361668475526303, 9.483093773043967448584297559871

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