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Properties

Label 2-925-185.84-c1-0-41
Degree $2$
Conductor $925$
Sign $-0.695 + 0.718i$
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  + (0.479 + 0.276i)2-s + (−1.80 + 1.04i)3-s + (−0.846 − 1.46i)4-s − 1.15·6-s + (3.61 − 2.08i)7-s − 2.04i·8-s + (0.670 − 1.16i)9-s − 3.31·11-s + (3.05 + 1.76i)12-s + (−0.365 + 0.211i)13-s + 2.30·14-s + (−1.12 + 1.95i)16-s + (−1.57 − 0.909i)17-s + (0.643 − 0.371i)18-s + (1.28 + 2.22i)19-s + ⋯
L(s)  = 1  + (0.339 + 0.195i)2-s + (−1.04 + 0.601i)3-s + (−0.423 − 0.733i)4-s − 0.470·6-s + (1.36 − 0.787i)7-s − 0.723i·8-s + (0.223 − 0.387i)9-s − 0.998·11-s + (0.882 + 0.509i)12-s + (−0.101 + 0.0585i)13-s + 0.617·14-s + (−0.281 + 0.488i)16-s + (−0.381 − 0.220i)17-s + (0.151 − 0.0875i)18-s + (0.294 + 0.509i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.695 + 0.718i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188907 - 0.445974i\)
\(L(\frac12)\) \(\approx\) \(0.188907 - 0.445974i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.66 + 5.46i)T \)
good2 \( 1 + (-0.479 - 0.276i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.80 - 1.04i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.61 + 2.08i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.31T + 11T^{2} \)
13 \( 1 + (0.365 - 0.211i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.57 + 0.909i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.28 - 2.22i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.981iT - 23T^{2} \)
29 \( 1 + 7.35T + 29T^{2} \)
31 \( 1 + 6.58T + 31T^{2} \)
41 \( 1 + (3.14 + 5.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.10iT - 43T^{2} \)
47 \( 1 + 10.6iT - 47T^{2} \)
53 \( 1 + (5.40 + 3.11i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.556 - 0.963i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.40 - 5.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.05 - 3.49i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.43 + 11.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.36iT - 73T^{2} \)
79 \( 1 + (7.18 + 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.959 - 0.554i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.18 - 5.50i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.47iT - 97T^{2} \)
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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

−10.07240462373176751248307408955, −9.093830739489412661176568600601, −7.933633497374524308048530286696, −7.16115349324485945377978663536, −5.87102306036800518597275063625, −5.28298836890663321424631190156, −4.69822764534287145986326359729, −3.89052712042736134401839715811, −1.80752154997081209192478136293, −0.23089260168425941501395497407, 1.75286117147423500320616498912, 2.93447318068431903532784978546, 4.41022909400465904746526302053, 5.26829240176490600113347804987, 5.69069909496555550400400878910, 7.09451261671299230643189171192, 7.83363848950133353243581820282, 8.537722738960426304592021254372, 9.444651969335625884683055852983, 10.95089478727242382575780528856

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