Properties

Label 2-70e2-5.4-c1-0-12
Degree $2$
Conductor $4900$
Sign $-0.894 - 0.447i$
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  + 2i·3-s − 9-s + 3·11-s + 4i·13-s + 2·19-s − 3i·23-s + 4i·27-s − 9·29-s − 8·31-s + 6i·33-s − 5i·37-s − 8·39-s + 6·41-s + 11i·43-s + 6i·47-s + ⋯
L(s)  = 1  + 1.15i·3-s − 0.333·9-s + 0.904·11-s + 1.10i·13-s + 0.458·19-s − 0.625i·23-s + 0.769i·27-s − 1.67·29-s − 1.43·31-s + 1.04i·33-s − 0.821i·37-s − 1.28·39-s + 0.937·41-s + 1.67i·43-s + 0.875i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 4900,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.630736842\)
\(L(\frac12)\) \(\approx\) \(1.630736842\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 3T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 5iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 11iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 7T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.946003870512646826768909572706, −7.82693856811984693533381047062, −7.12651668971315294136418562459, −6.36140337438827289835667728153, −5.54064799976285830487855539950, −4.75763976041528938941185958969, −3.96932067493323880111263117415, −3.67360931563469139047896521133, −2.38231624300794314603861138876, −1.32049414643531234638633372558, 0.45392276574366017408121681965, 1.47535798637462943977854971429, 2.21681233367594693491822619975, 3.40422041411587725103701471552, 4.00365918262057990918171594480, 5.40375772514510427236820358220, 5.67667741574166284614229752916, 6.80989579169979232757845307786, 7.12217062758546627375529513905, 7.86194639426013450934077220128

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