Properties

Label 2-70e2-5.4-c1-0-24
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  + i·3-s + 2·9-s + 3·11-s i·13-s + 3i·17-s + 2·19-s + 6i·23-s + 5i·27-s + 9·29-s − 8·31-s + 3i·33-s − 10i·37-s + 39-s − 2i·43-s + 3i·47-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.666·9-s + 0.904·11-s − 0.277i·13-s + 0.727i·17-s + 0.458·19-s + 1.25i·23-s + 0.962i·27-s + 1.67·29-s − 1.43·31-s + 0.522i·33-s − 1.64i·37-s + 0.160·39-s − 0.304i·43-s + 0.437i·47-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

\(L(1)\) \(\approx\) \(2.272284336\)
\(L(\frac12)\) \(\approx\) \(2.272284336\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 3T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 17iT - 97T^{2} \)
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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

−8.603065637268299122701240434488, −7.46597890896270320179846014565, −7.14171544229647254250842592286, −6.10849222096789831357377938109, −5.48197429969527756623327698934, −4.58421121154922232442410868444, −3.86818284312384878420606600200, −3.31651143966793661020085412698, −1.98689270840773951893061333010, −1.06127112404737090895035348434, 0.74354561630153336456397729461, 1.62176828910906757940563330815, 2.60884281419296112573075834363, 3.58985731513789591328571091730, 4.49710170317929390161925092859, 5.07036069492517075456504060293, 6.33546966474839103338400339414, 6.62033696800988169340629896352, 7.32974718307803183203203355325, 8.067752254235522766690821442662

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