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Properties

Label 2-70e2-5.4-c1-0-27
Degree $2$
Conductor $4900$
Sign $-0.447 - 0.894i$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
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  + 2.41i·3-s − 2.82·9-s + 1.82·11-s + 6.41i·13-s − 3.58i·17-s + 7.65·19-s + 3.41i·23-s + 0.414i·27-s + 4.65·29-s + 7.41·31-s + 4.41i·33-s + 0.585i·37-s − 15.4·39-s + 3.41·41-s + 0.343i·43-s + ⋯
L(s)  = 1  + 1.39i·3-s − 0.942·9-s + 0.551·11-s + 1.77i·13-s − 0.869i·17-s + 1.75·19-s + 0.711i·23-s + 0.0797i·27-s + 0.864·29-s + 1.33·31-s + 0.768i·33-s + 0.0963i·37-s − 2.47·39-s + 0.533·41-s + 0.0523i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(39.1266\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4900} (2549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.246852013\)
\(L(\frac12)\) \(\approx\) \(2.246852013\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3 \( 1 - 2.41iT - 3T^{2} \)
11 \( 1 - 1.82T + 11T^{2} \)
13 \( 1 - 6.41iT - 13T^{2} \)
17 \( 1 + 3.58iT - 17T^{2} \)
19 \( 1 - 7.65T + 19T^{2} \)
23 \( 1 - 3.41iT - 23T^{2} \)
29 \( 1 - 4.65T + 29T^{2} \)
31 \( 1 - 7.41T + 31T^{2} \)
37 \( 1 - 0.585iT - 37T^{2} \)
41 \( 1 - 3.41T + 41T^{2} \)
43 \( 1 - 0.343iT - 43T^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + 12.2iT - 53T^{2} \)
59 \( 1 + 0.585T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + 3.07iT - 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 - 9.72iT - 97T^{2} \)
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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

−8.893484449005863514537185191357, −7.80682736677298438023937372796, −6.95941054499430448311530361611, −6.38187474100163801327050614688, −5.16737017349073616703701956798, −4.90873737227662983601825081132, −3.95756469741968028386452128194, −3.47227237006939469496673212468, −2.39922619072475170118295870180, −1.10568913833045997911908541267, 0.78296298390531300076142340889, 1.28603940258140992574964638892, 2.59767516013113999166936826792, 3.14152472128562187555756998001, 4.30920553880049508166103750721, 5.31841161114391276987868350994, 6.07174600949835282999908399347, 6.52759582392083547785143883541, 7.49464457244106326482292869278, 7.85702214714817801290571005471

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