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Properties

Label 2-925-185.174-c1-0-2
Degree $2$
Conductor $925$
Sign $-0.567 - 0.823i$
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  + (−1.50 + 0.868i)2-s + (−1.10 − 0.639i)3-s + (0.509 − 0.882i)4-s + 2.22·6-s + (−1.52 − 0.878i)7-s − 1.70i·8-s + (−0.682 − 1.18i)9-s + 0.0277·11-s + (−1.12 + 0.651i)12-s + (−3.09 − 1.78i)13-s + 3.05·14-s + (2.49 + 4.32i)16-s + (−1.69 + 0.978i)17-s + (2.05 + 1.18i)18-s + (0.650 − 1.12i)19-s + ⋯
L(s)  = 1  + (−1.06 + 0.614i)2-s + (−0.639 − 0.369i)3-s + (0.254 − 0.441i)4-s + 0.906·6-s + (−0.575 − 0.331i)7-s − 0.602i·8-s + (−0.227 − 0.394i)9-s + 0.00837·11-s + (−0.325 + 0.188i)12-s + (−0.859 − 0.496i)13-s + 0.815·14-s + (0.624 + 1.08i)16-s + (−0.411 + 0.237i)17-s + (0.484 + 0.279i)18-s + (0.149 − 0.258i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.567 - 0.823i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.567 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0870787 + 0.165890i\)
\(L(\frac12)\) \(\approx\) \(0.0870787 + 0.165890i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.70 - 2.12i)T \)
good2 \( 1 + (1.50 - 0.868i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.10 + 0.639i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.52 + 0.878i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 0.0277T + 11T^{2} \)
13 \( 1 + (3.09 + 1.78i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.69 - 0.978i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.650 + 1.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.78iT - 23T^{2} \)
29 \( 1 - 2.02T + 29T^{2} \)
31 \( 1 - 0.467T + 31T^{2} \)
41 \( 1 + (1.35 - 2.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 0.411iT - 43T^{2} \)
47 \( 1 - 0.517iT - 47T^{2} \)
53 \( 1 + (3.16 - 1.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.75 + 3.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.72 - 9.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.7 - 7.96i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.46 - 2.54i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.0iT - 73T^{2} \)
79 \( 1 + (1.70 - 2.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.1 - 5.88i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.108 + 0.187i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.47iT - 97T^{2} \)
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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

−10.08673672304901839359691120642, −9.529312816363726331453040162821, −8.628239638139853087292814717780, −7.83064358727477922275542345726, −6.88221981119437608617008252298, −6.50967106942907296739222497146, −5.47849400639961266083874857961, −4.16828308674417050447183682105, −2.88048179622085478548876024197, −0.943082641399321020808200290010, 0.17658983201630056166012240047, 1.94579991293560925665636045349, 2.97152413579416719111736783932, 4.55275959133584067560336464227, 5.37649471632909382280883712891, 6.30776397720429868158912125776, 7.48537061030920662822571812197, 8.314925322649931987951222721261, 9.352361248149355592059177179348, 9.688008547898407125152892378448

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