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Properties

Label 2-925-185.84-c1-0-24
Degree $2$
Conductor $925$
Sign $0.136 + 0.990i$
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  + (−2.05 − 1.18i)2-s + (−0.0643 + 0.0371i)3-s + (1.82 + 3.15i)4-s + 0.176·6-s + (3.20 − 1.84i)7-s − 3.90i·8-s + (−1.49 + 2.59i)9-s − 1.25·11-s + (−0.234 − 0.135i)12-s + (4.24 − 2.44i)13-s − 8.78·14-s + (−0.998 + 1.73i)16-s + (−5.02 − 2.90i)17-s + (6.16 − 3.55i)18-s + (−0.0552 − 0.0957i)19-s + ⋯
L(s)  = 1  + (−1.45 − 0.840i)2-s + (−0.0371 + 0.0214i)3-s + (0.911 + 1.57i)4-s + 0.0720·6-s + (1.21 − 0.698i)7-s − 1.38i·8-s + (−0.499 + 0.864i)9-s − 0.378·11-s + (−0.0676 − 0.0390i)12-s + (1.17 − 0.679i)13-s − 2.34·14-s + (−0.249 + 0.432i)16-s + (−1.21 − 0.703i)17-s + (1.45 − 0.838i)18-s + (−0.0126 − 0.0219i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.136 + 0.990i$
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: \((2,\ 925,\ (\ :1/2),\ 0.136 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.602694 - 0.525288i\)
\(L(\frac12)\) \(\approx\) \(0.602694 - 0.525288i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 + 1.18i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.0643 - 0.0371i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.20 + 1.84i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 1.25T + 11T^{2} \)
13 \( 1 + (-4.24 + 2.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.02 + 2.90i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0552 + 0.0957i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.96iT - 23T^{2} \)
29 \( 1 - 9.26T + 29T^{2} \)
31 \( 1 - 2.89T + 31T^{2} \)
41 \( 1 + (-2.81 - 4.86i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 0.220iT - 43T^{2} \)
47 \( 1 + 0.197iT - 47T^{2} \)
53 \( 1 + (-6.23 - 3.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.77 + 9.99i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.16 - 2.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.7 + 7.36i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.33 - 5.78i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.55iT - 73T^{2} \)
79 \( 1 + (6.15 + 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.49 + 4.90i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.02 + 8.70i)T + (-44.5 - 77.0i)T^{2} \)
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

−10.11891317807277210770104096341, −8.945005495760515157743803163268, −8.270162134034417820912955746420, −7.88109619670594241948017405744, −6.89336396749481985807965111493, −5.42603275809519085860996442026, −4.41935851586381913208732057254, −2.98688186965585279516298190068, −1.97048219412522010721393803867, −0.77558151818361216161577956765, 1.08848121227594146627432303721, 2.33125254590356583462478216581, 4.13836214676897739543862506563, 5.40142473822017421666411721827, 6.35699452850146704069081753588, 6.80713276715389935141810572933, 8.243325323382623028154907446324, 8.498432536469164050978821825943, 8.976501446005639879152409019882, 10.09600067482174540937969602972

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