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Properties

Label 2-725-29.28-c1-0-38
Degree $2$
Conductor $725$
Sign $-0.356 + 0.934i$
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  + 0.285i·2-s − 3.21i·3-s + 1.91·4-s + 0.918·6-s + 2.91·7-s + 1.11i·8-s − 7.35·9-s − 3.21i·11-s − 6.17i·12-s − 4.35·13-s + 0.833i·14-s + 3.51·16-s − 1.97i·17-s − 2.09i·18-s − 1.40i·19-s + ⋯
L(s)  = 1  + 0.201i·2-s − 1.85i·3-s + 0.959·4-s + 0.374·6-s + 1.10·7-s + 0.395i·8-s − 2.45·9-s − 0.970i·11-s − 1.78i·12-s − 1.20·13-s + 0.222i·14-s + 0.879·16-s − 0.478i·17-s − 0.494i·18-s − 0.322i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.356 + 0.934i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.356 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08620 - 1.57662i\)
\(L(\frac12)\) \(\approx\) \(1.08620 - 1.57662i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.91 + 5.03i)T \)
good2 \( 1 - 0.285iT - 2T^{2} \)
3 \( 1 + 3.21iT - 3T^{2} \)
7 \( 1 - 2.91T + 7T^{2} \)
11 \( 1 + 3.21iT - 11T^{2} \)
13 \( 1 + 4.35T + 13T^{2} \)
17 \( 1 + 1.97iT - 17T^{2} \)
19 \( 1 + 1.40iT - 19T^{2} \)
23 \( 1 - 3.43T + 23T^{2} \)
31 \( 1 - 8.98iT - 31T^{2} \)
37 \( 1 - 4.46iT - 37T^{2} \)
41 \( 1 + 9.39iT - 41T^{2} \)
43 \( 1 + 6.84iT - 43T^{2} \)
47 \( 1 - 3.21iT - 47T^{2} \)
53 \( 1 - 6.19T + 53T^{2} \)
59 \( 1 - 2.35T + 59T^{2} \)
61 \( 1 - 8.24iT - 61T^{2} \)
67 \( 1 - 0.563T + 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 - 6.84iT - 79T^{2} \)
83 \( 1 - 1.08T + 83T^{2} \)
89 \( 1 - 3.62iT - 89T^{2} \)
97 \( 1 + 3.78iT - 97T^{2} \)
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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

−10.45979242806816669400192154483, −8.757897301607774110622012596817, −8.189177134507152193412241456121, −7.24476095358671001258235714879, −7.00412741451218788026561187850, −5.86606715707934208147217461811, −5.10949141952006439406481211640, −2.94093349895643344499290010514, −2.16022695594923272511651351240, −1.00675605031382493785599873009, 2.05782556321539715209785382786, 3.16530465229857772103149034885, 4.40039843246167996475859184848, 4.94219734967353611114567390738, 5.97481386033171975570265221311, 7.32713871166914271915427312266, 8.145311553341745154090780853260, 9.288248419665370146775526025394, 9.990412766688595988341295777909, 10.57110242423971819061734526375

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