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Properties

Label 2-725-145.128-c1-0-26
Degree $2$
Conductor $725$
Sign $0.934 - 0.355i$
Analytic cond. $5.78915$
Root an. cond. $2.40606$
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  + 2.23·2-s + 3.00·4-s + (−2.23 + 2.23i)7-s + 2.23·8-s + 3·9-s + (4 + 4i)11-s + (4.47 − 4.47i)13-s + (−5.00 + 5.00i)14-s − 0.999·16-s − 4.47·17-s + 6.70·18-s + (2 − 2i)19-s + (8.94 + 8.94i)22-s + (2.23 + 2.23i)23-s + (10.0 − 10.0i)26-s + ⋯
L(s)  = 1  + 1.58·2-s + 1.50·4-s + (−0.845 + 0.845i)7-s + 0.790·8-s + 9-s + (1.20 + 1.20i)11-s + (1.24 − 1.24i)13-s + (−1.33 + 1.33i)14-s − 0.249·16-s − 1.08·17-s + 1.58·18-s + (0.458 − 0.458i)19-s + (1.90 + 1.90i)22-s + (0.466 + 0.466i)23-s + (1.96 − 1.96i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.934 - 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.59874 + 0.661037i\)
\(L(\frac12)\) \(\approx\) \(3.59874 + 0.661037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 + (2 - 5i)T \)
good2 \( 1 - 2.23T + 2T^{2} \)
3 \( 1 - 3T^{2} \)
7 \( 1 + (2.23 - 2.23i)T - 7iT^{2} \)
11 \( 1 + (-4 - 4i)T + 11iT^{2} \)
13 \( 1 + (-4.47 + 4.47i)T - 13iT^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 + (-2 + 2i)T - 19iT^{2} \)
23 \( 1 + (-2.23 - 2.23i)T + 23iT^{2} \)
31 \( 1 + (4 + 4i)T + 31iT^{2} \)
37 \( 1 + 4.47iT - 37T^{2} \)
41 \( 1 + (-1 + i)T - 41iT^{2} \)
43 \( 1 + 4.47iT - 43T^{2} \)
47 \( 1 - 4.47iT - 47T^{2} \)
53 \( 1 + (8.94 + 8.94i)T + 53iT^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (9 + 9i)T + 61iT^{2} \)
67 \( 1 + (2.23 + 2.23i)T + 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + 4.47T + 73T^{2} \)
79 \( 1 + (-2 + 2i)T - 79iT^{2} \)
83 \( 1 + (6.70 + 6.70i)T + 83iT^{2} \)
89 \( 1 + (-7 + 7i)T - 89iT^{2} \)
97 \( 1 - 4.47iT - 97T^{2} \)
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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

−10.75392731943724428366924823708, −9.489310946056067993443168997642, −9.000070354577974621711271143158, −7.36810030931368005630144869504, −6.63513986018119108592325641281, −5.91790464333501754952404888116, −4.93474394528426623314835201522, −3.96434248664853086030964984420, −3.20339209386446897602257084474, −1.84945081486006008176496898445, 1.45656544434698960730519338340, 3.26401193636293725225566517783, 3.95760664875555329702150104354, 4.50129014447432163138427666023, 6.11531943848649549094240119005, 6.45670073165580359800912194663, 7.19106774701321637461178132177, 8.771676933584255895763840582996, 9.477997937415862479166578419093, 10.76042422674138332150683148058

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