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Properties

Label 2-925-5.4-c1-0-3
Degree $2$
Conductor $925$
Sign $0.894 + 0.447i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
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  + 2.72i·2-s + 2.29i·3-s − 5.41·4-s − 6.24·6-s + 3.82i·7-s − 9.30i·8-s − 2.25·9-s − 4.41·11-s − 12.4i·12-s + 3.67i·13-s − 10.4·14-s + 14.5·16-s − 2.28i·17-s − 6.14i·18-s + 2.39·19-s + ⋯
L(s)  = 1  + 1.92i·2-s + 1.32i·3-s − 2.70·4-s − 2.54·6-s + 1.44i·7-s − 3.29i·8-s − 0.752·9-s − 1.33·11-s − 3.58i·12-s + 1.01i·13-s − 2.78·14-s + 3.62·16-s − 0.554i·17-s − 1.44i·18-s + 0.548·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821684 - 0.193973i\)
\(L(\frac12)\) \(\approx\) \(0.821684 - 0.193973i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 - iT \)
good2 \( 1 - 2.72iT - 2T^{2} \)
3 \( 1 - 2.29iT - 3T^{2} \)
7 \( 1 - 3.82iT - 7T^{2} \)
11 \( 1 + 4.41T + 11T^{2} \)
13 \( 1 - 3.67iT - 13T^{2} \)
17 \( 1 + 2.28iT - 17T^{2} \)
19 \( 1 - 2.39T + 19T^{2} \)
23 \( 1 - 0.265iT - 23T^{2} \)
29 \( 1 - 6.58T + 29T^{2} \)
31 \( 1 - 2.34T + 31T^{2} \)
41 \( 1 + 4.41T + 41T^{2} \)
43 \( 1 + 7.71iT - 43T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 - 0.109iT - 53T^{2} \)
59 \( 1 - 2.00T + 59T^{2} \)
61 \( 1 - 3.96T + 61T^{2} \)
67 \( 1 - 6.80iT - 67T^{2} \)
71 \( 1 + 5.79T + 71T^{2} \)
73 \( 1 - 0.140iT - 73T^{2} \)
79 \( 1 - 6.62T + 79T^{2} \)
83 \( 1 + 13.9iT - 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + 8.94iT - 97T^{2} \)
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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

−10.31892971211019705342679962177, −9.661383485057328941945804107749, −8.966765607356632013032481307155, −8.455043728017568169232643743406, −7.47621647508788617498020686339, −6.46091643411655891206517040749, −5.49072049436400653790090384087, −5.05429571463933879766379714700, −4.29084205143610601863771950725, −2.93250636459687595591788338551, 0.44796225027681243930166302800, 1.29673367807400129306463782690, 2.51078558619050788589290600863, 3.37677970202854347661616109158, 4.51771145598034684098831062759, 5.51007114512561863550633796083, 6.94374080227281110587386499589, 8.030969592806617863009882975917, 8.239168096882278608465822444910, 9.829694210046382603898277956703

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