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Properties

Label 2-1450-145.128-c1-0-2
Degree $2$
Conductor $1450$
Sign $-0.910 - 0.412i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
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-s + 0.496i·3-s + 4-s − 0.496i·6-s + (−0.303 + 0.303i)7-s − 8-s + 2.75·9-s + (−0.496 − 0.496i)11-s + 0.496i·12-s + (−1.40 + 1.40i)13-s + (0.303 − 0.303i)14-s + 16-s − 4.95·17-s − 2.75·18-s + (−5.45 + 5.45i)19-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.286i·3-s + 0.5·4-s − 0.202i·6-s + (−0.114 + 0.114i)7-s − 0.353·8-s + 0.917·9-s + (−0.149 − 0.149i)11-s + 0.143i·12-s + (−0.389 + 0.389i)13-s + (0.0811 − 0.0811i)14-s + 0.250·16-s − 1.20·17-s − 0.648·18-s + (−1.25 + 1.25i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $-0.910 - 0.412i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (1143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ -0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4483778589\)
\(L(\frac12)\) \(\approx\) \(0.4483778589\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
29 \( 1 + (-4.95 + 2.10i)T \)
good3 \( 1 - 0.496iT - 3T^{2} \)
7 \( 1 + (0.303 - 0.303i)T - 7iT^{2} \)
11 \( 1 + (0.496 + 0.496i)T + 11iT^{2} \)
13 \( 1 + (1.40 - 1.40i)T - 13iT^{2} \)
17 \( 1 + 4.95T + 17T^{2} \)
19 \( 1 + (5.45 - 5.45i)T - 19iT^{2} \)
23 \( 1 + (2.40 + 2.40i)T + 23iT^{2} \)
31 \( 1 + (3.50 + 3.50i)T + 31iT^{2} \)
37 \( 1 - 8.31iT - 37T^{2} \)
41 \( 1 + (-2.81 + 2.81i)T - 41iT^{2} \)
43 \( 1 + 3.26iT - 43T^{2} \)
47 \( 1 - 12.5iT - 47T^{2} \)
53 \( 1 + (4.04 + 4.04i)T + 53iT^{2} \)
59 \( 1 - 4.09iT - 59T^{2} \)
61 \( 1 + (3.50 + 3.50i)T + 61iT^{2} \)
67 \( 1 + (8.60 + 8.60i)T + 67iT^{2} \)
71 \( 1 - 2.71iT - 71T^{2} \)
73 \( 1 + 6.94T + 73T^{2} \)
79 \( 1 + (3.51 - 3.51i)T - 79iT^{2} \)
83 \( 1 + (-0.452 - 0.452i)T + 83iT^{2} \)
89 \( 1 + (5.64 - 5.64i)T - 89iT^{2} \)
97 \( 1 - 18.7iT - 97T^{2} \)
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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

−9.833492641160757410617316793333, −9.149820321347902283585323148224, −8.319109727195955895014448454597, −7.60986387079901852405187156762, −6.60175743983000607415270188221, −6.08232014427385878126221941573, −4.64779473126432060135025085798, −4.03283016475273753807458309799, −2.60444086325433693929473464699, −1.60571979826166025427554255041, 0.21603655682699383450129297324, 1.74539484306845888020368584878, 2.66623073613813870517835985493, 4.06959316191416465574906645715, 4.93161810001203971792385818514, 6.19245608551745363267342577667, 6.98096520262748612422469519068, 7.42317804784371752589580696816, 8.521027199946350809508620832214, 9.059831028705079520013443945229

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