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Properties

Label 2-70e2-5.4-c1-0-10
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  + 0.414i·3-s + 2.82·9-s − 3.82·11-s − 3.58i·13-s + 6.41i·17-s + 3.65·19-s + 0.585i·23-s + 2.41i·27-s − 6.65·29-s − 4.58·31-s − 1.58i·33-s + 3.41i·37-s + 1.48·39-s − 0.585·41-s + 11.6i·43-s + ⋯
L(s)  = 1  + 0.239i·3-s + 0.942·9-s − 1.15·11-s − 0.994i·13-s + 1.55i·17-s + 0.838·19-s + 0.122i·23-s + 0.464i·27-s − 1.23·29-s − 0.823·31-s − 0.276i·33-s + 0.561i·37-s + 0.237·39-s − 0.0914·41-s + 1.77i·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

\(L(1)\) \(\approx\) \(1.179959748\)
\(L(\frac12)\) \(\approx\) \(1.179959748\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.414iT - 3T^{2} \)
11 \( 1 + 3.82T + 11T^{2} \)
13 \( 1 + 3.58iT - 13T^{2} \)
17 \( 1 - 6.41iT - 17T^{2} \)
19 \( 1 - 3.65T + 19T^{2} \)
23 \( 1 - 0.585iT - 23T^{2} \)
29 \( 1 + 6.65T + 29T^{2} \)
31 \( 1 + 4.58T + 31T^{2} \)
37 \( 1 - 3.41iT - 37T^{2} \)
41 \( 1 + 0.585T + 41T^{2} \)
43 \( 1 - 11.6iT - 43T^{2} \)
47 \( 1 + 8.89iT - 47T^{2} \)
53 \( 1 + 3.75iT - 53T^{2} \)
59 \( 1 - 3.41T + 59T^{2} \)
61 \( 1 + 5.17T + 61T^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 - 6.48T + 71T^{2} \)
73 \( 1 + 5.17iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 - 15.7iT - 97T^{2} \)
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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

−8.315581720118524894071453582848, −7.79466697533798021639431825953, −7.23532908759563021828917965778, −6.24700802632686831153231807168, −5.46927468241377571161030697524, −4.95079689389561734618318894430, −3.89215987766809778362626856643, −3.31696723610590771506214467688, −2.20812345610766937435832434637, −1.20037280523249234654768554575, 0.32497950065969216262147096716, 1.64415457384722035998113925618, 2.46674537457991338181424707426, 3.44950657669313724282726368188, 4.38053296214254590113932364587, 5.09588369239581722565510151940, 5.75461101367863306879808312747, 6.82250348433490509806879225946, 7.42248012906629211928394680415, 7.66655454201292643408065147153

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