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Properties

Label 1-185-185.147-r1-0-0
Degree $1$
Conductor $185$
Sign $0.525 - 0.850i$
Analytic cond. $19.8810$
Root an. cond. $19.8810$
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  − i·2-s − i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s + 11-s + i·12-s + i·13-s + 14-s + 16-s − i·17-s + i·18-s + 19-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s + 11-s + i·12-s + i·13-s + 14-s + 16-s − i·17-s + i·18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.362745055 - 0.7597801299i\)
\(L(\frac12)\) \(\approx\) \(1.362745055 - 0.7597801299i\)
\(L(1)\) \(\approx\) \(0.8786791583 - 0.5430535850i\)
\(L(1)\) \(\approx\) \(0.8786791583 - 0.5430535850i\)

Euler product

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

−27.017148951914119539404808491489, −26.265987935102956027162030300150, −25.35321694084293579079035294310, −24.358844427320842226272935590851, −23.18933586724123631429828339487, −22.5174834164435845948051090907, −21.69924998070841759267066786929, −20.34878883776893627425115795224, −19.5916706594339120207591082638, −17.97349607220108483179706191923, −17.08157396035632506287031533499, −16.475960050476777270701734514897, −15.418094972161947060030740447370, −14.526041664954588827157580356833, −13.78351619302356768492964451784, −12.41903541474808657514871245322, −10.7803766233085925070181244312, −9.97671424786873002197546110014, −8.88871815883479013945995938416, −7.854363910132304386293792319739, −6.57862214049196607160757972147, −5.44286072323083786461810932813, −4.270812246195528542689799012819, −3.46691219785239043522439856767, −0.70303455173389765535699632588, 1.100049881807020823066915602387, 2.183066958785714354892624780953, 3.33677932802236617378974808625, 4.98499739913981384152391683210, 6.194209950835713431705260704835, 7.56887153864626757446870075194, 8.942186240601718573994665669, 9.472106110582466198381854886636, 11.48125145192271152113067295287, 11.707024127694018475730230856883, 12.729912760303820931233554029021, 13.883888370519717140519181578515, 14.51174970486684130422250536210, 16.2631704158450322736011911860, 17.63043365035986641159867517291, 18.2631235528282940755247929153, 19.20973658506300672509340751473, 19.80389756459992820805329138833, 21.030186148368487991467426461379, 22.06187448904356502403529311621, 22.79877244292700658324515398717, 23.93252380628729716541239873422, 24.85079691761213425192372896669, 25.80254630935657295690502950376, 27.094883179165924095938004661484

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