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Properties

Label 1-1925-1925.1203-r0-0-0
Degree $1$
Conductor $1925$
Sign $-0.0390 - 0.999i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.951 − 0.309i)3-s + (0.809 + 0.587i)4-s + (0.809 + 0.587i)6-s + (−0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (−0.587 − 0.809i)12-s + (−0.587 + 0.809i)13-s + (0.309 + 0.951i)16-s + (0.587 − 0.809i)17-s + (−0.587 − 0.809i)18-s + (−0.809 + 0.587i)19-s + (0.587 − 0.809i)23-s + (0.309 + 0.951i)24-s + (0.809 − 0.587i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.951 − 0.309i)3-s + (0.809 + 0.587i)4-s + (0.809 + 0.587i)6-s + (−0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (−0.587 − 0.809i)12-s + (−0.587 + 0.809i)13-s + (0.309 + 0.951i)16-s + (0.587 − 0.809i)17-s + (−0.587 − 0.809i)18-s + (−0.809 + 0.587i)19-s + (0.587 − 0.809i)23-s + (0.309 + 0.951i)24-s + (0.809 − 0.587i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0390 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0390 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.0390 - 0.999i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ -0.0390 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3291125900 - 0.3422247815i\)
\(L(\frac12)\) \(\approx\) \(0.3291125900 - 0.3422247815i\)
\(L(1)\) \(\approx\) \(0.4807334683 - 0.1103221000i\)
\(L(1)\) \(\approx\) \(0.4807334683 - 0.1103221000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 - T \)
43 \( 1 + iT \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 + T \)
73 \( 1 - iT \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.09459717232867897137750409630, −19.28597089839793551515997877606, −18.70669807663406420797803294991, −17.78784252496485018338295885465, −17.20696613845148755615410605544, −16.927889439007517090995351142451, −15.91740021776576695341508410082, −15.2427547460411779057359732571, −14.87219493504276332098530858831, −13.57757378602733602523667121451, −12.50971151775356240167537603434, −11.977120338474907749400787599272, −11.02772909947312871031714467969, −10.46066421099546040845425680156, −9.92004830268327326910461889042, −9.017033372426162825509232461797, −8.205312449740779977163413295986, −7.2387406007646689486565756828, −6.704407915958492826943901295986, −5.63615972600379271128291341419, −5.30742184719502793546700740448, −4.09966061653332688021179805541, −2.96533077220735899119692642422, −1.79014249094529390261415548536, −0.79781941566223598492242796938, 0.35954752738641735978936564081, 1.49373859103514757196167003098, 2.22928629249808430978706325545, 3.36108928008295435017147400025, 4.497243247668356933835560998, 5.34781687956030816948519926662, 6.47943845425044882158487857232, 6.90914295960191824267731869658, 7.7103803193425524529082985048, 8.59920957363577402863262446044, 9.49243142502307224581972330706, 10.20986791920400580394243802441, 10.905780442186601554265702245437, 11.56496596656730771866115922693, 12.39719200926111433812374643363, 12.66692356571837818343919173217, 13.93486530745990820607711505012, 14.82294135039437805165603256475, 15.88127716251042262634331819450, 16.518410562894116177389650475, 16.920564515322577318876278097176, 17.71276855260250755779711830217, 18.42975670859583593727336034556, 18.96990136679471285993636457063, 19.54301828044105766995361238665

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