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Properties

Label 2-2450-5.4-c1-0-49
Degree $2$
Conductor $2450$
Sign $0.894 + 0.447i$
Analytic cond. $19.5633$
Root an. cond. $4.42304$
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·2-s − 4-s − i·8-s + 3·9-s + 4·11-s − 6i·13-s + 16-s − 2i·17-s + 3i·18-s + 4i·22-s + 6·26-s − 6·29-s − 8·31-s + i·32-s + 2·34-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.353i·8-s + 9-s + 1.20·11-s − 1.66i·13-s + 0.250·16-s − 0.485i·17-s + 0.707i·18-s + 0.852i·22-s + 1.17·26-s − 1.11·29-s − 1.43·31-s + 0.176i·32-s + 0.342·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(19.5633\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2450} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.692880227\)
\(L(\frac12)\) \(\approx\) \(1.692880227\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - 3T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.989726762343082665394647968083, −7.88699348270199839331785260553, −7.35723255013866401975578666122, −6.70621357434304784754136437266, −5.72049455523012331634926904893, −5.15400445526562305421510023393, −4.01466257195231029512624797866, −3.46279013586838455807860304301, −1.90579381612495718147824086067, −0.60619432205379762246235719692, 1.39990125129625124263851426966, 1.90173568409539422613977127687, 3.40127290978770289913697773548, 4.14732215425750944075815496081, 4.66023702078066563583429189332, 5.93812612562973689655658734544, 6.75455176798058391188704523467, 7.36346163818633471784470462890, 8.474702556584352627174212379876, 9.268353699802693065051488763483

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