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Properties

Label 2-725-145.4-c1-0-13
Degree $2$
Conductor $725$
Sign $-0.282 - 0.959i$
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  + (−0.0209 + 0.0919i)2-s + (−0.209 + 0.100i)3-s + (1.79 + 0.863i)4-s + (−0.00487 − 0.0213i)6-s + (1.49 + 3.10i)7-s + (−0.234 + 0.294i)8-s + (−1.83 + 2.30i)9-s + (−0.00755 + 0.00602i)11-s − 0.462·12-s + (−3.63 + 2.90i)13-s + (−0.317 + 0.0724i)14-s + (2.46 + 3.08i)16-s − 5.88·17-s + (−0.173 − 0.217i)18-s + (2.32 − 4.83i)19-s + ⋯
L(s)  = 1  + (−0.0148 + 0.0650i)2-s + (−0.120 + 0.0581i)3-s + (0.896 + 0.431i)4-s + (−0.00198 − 0.00871i)6-s + (0.565 + 1.17i)7-s + (−0.0830 + 0.104i)8-s + (−0.612 + 0.767i)9-s + (−0.00227 + 0.00181i)11-s − 0.133·12-s + (−1.00 + 0.804i)13-s + (−0.0847 + 0.0193i)14-s + (0.615 + 0.771i)16-s − 1.42·17-s + (−0.0408 − 0.0512i)18-s + (0.534 − 1.10i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.282 - 0.959i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.282 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.915066 + 1.22325i\)
\(L(\frac12)\) \(\approx\) \(0.915066 + 1.22325i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.15 + 3.42i)T \)
good2 \( 1 + (0.0209 - 0.0919i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.209 - 0.100i)T + (1.87 - 2.34i)T^{2} \)
7 \( 1 + (-1.49 - 3.10i)T + (-4.36 + 5.47i)T^{2} \)
11 \( 1 + (0.00755 - 0.00602i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (3.63 - 2.90i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + 5.88T + 17T^{2} \)
19 \( 1 + (-2.32 + 4.83i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (3.11 - 0.711i)T + (20.7 - 9.97i)T^{2} \)
31 \( 1 + (-9.42 - 2.15i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (-5.25 + 6.59i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 - 7.41iT - 41T^{2} \)
43 \( 1 + (0.472 + 2.07i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-2.03 - 2.55i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-9.49 - 2.16i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 + 0.911T + 59T^{2} \)
61 \( 1 + (2.97 + 6.17i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (-6.75 - 5.38i)T + (14.9 + 65.3i)T^{2} \)
71 \( 1 + (-9.02 - 11.3i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-2.73 - 11.9i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (-1.68 - 1.34i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (-3.03 + 6.29i)T + (-51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.908 + 0.207i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (-9.49 - 4.57i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.01519213709419817347582848314, −9.736157067809990619208270444970, −8.799190520334633496533240119958, −8.090528061941573916075934097740, −7.18143386966805075699415627037, −6.26937303586121423453913512346, −5.30774563570740405490945438369, −4.37725840163565032021370892175, −2.50715021594975290862967956856, −2.29282994002594392988890788163, 0.75813890015634340091317332260, 2.22876857193391934359201400931, 3.47904225288430816757783654644, 4.70314664850143999842160338835, 5.80414938661068692038317541993, 6.64237574883763051987610628775, 7.47244920186286680005719668809, 8.220234260911505101018824229993, 9.546512697223559762663952852666, 10.30508349935912230285897514494

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