Properties

Label 1-195-195.68-r0-0-0
Degree $1$
Conductor $195$
Sign $0.959 - 0.281i$
Analytic cond. $0.905576$
Root an. cond. $0.905576$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)7-s i·8-s + (0.5 − 0.866i)11-s + 14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s + (−0.866 − 0.5i)28-s + (−0.5 + 0.866i)29-s + 31-s + (0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)7-s i·8-s + (0.5 − 0.866i)11-s + 14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s + (−0.866 − 0.5i)28-s + (−0.5 + 0.866i)29-s + 31-s + (0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.959 - 0.281i$
Analytic conductor: \(0.905576\)
Root analytic conductor: \(0.905576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (0:\ ),\ 0.959 - 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7380609716 - 0.1060281823i\)
\(L(\frac12)\) \(\approx\) \(0.7380609716 - 0.1060281823i\)
\(L(1)\) \(\approx\) \(0.7200506066 - 0.1014099191i\)
\(L(1)\) \(\approx\) \(0.7200506066 - 0.1014099191i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + iT \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + iT \)
79 \( 1 - T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−26.828056134359104090739729692116, −26.15477484183996775318333625600, −25.31376984072046908682996371519, −24.46833024576671375167671668827, −23.24382574570490539291079758147, −22.688526767996809217396115767789, −21.11869513762016963877915814711, −19.9787418824012243075266056601, −19.4086592732294568144919673273, −18.34826360860243152095053385386, −17.23799974071064912474946976113, −16.63413132467952315955540626230, −15.51939700875446647799235285435, −14.66985834653685647332936093959, −13.44808340131484851702089351555, −12.18754689972420518683824667813, −10.910069240387918540213800516936, −9.8562171150608218372081031862, −9.20808467178953941139441700607, −7.77776832195723862040827138107, −6.913184737502084248287092587799, −5.93793251191954503122222403150, −4.428053053184624712112259218432, −2.73750900513436306530322114607, −1.0418016641120156393284815080, 1.11011687382691575410678988202, 2.81395148206164051328152603066, 3.64962862735336818598197176990, 5.65666621331659245081332515092, 6.82303398218657679367019932937, 8.02510179557561404776012226153, 9.14685144078286352149757633790, 9.83835877194517493476145402531, 11.08920676063131073288119847237, 12.03190508571776441579899504941, 12.91920593388761488619054944430, 14.21020795589167993163483090935, 15.72331345716894296447015615648, 16.42685316750209997153785941381, 17.33128504826037074996413476025, 18.76013073096563086768624748884, 18.96652935557913513609254619493, 20.14666869034656100794412007157, 21.13964420285661760065981122333, 22.03360978180242205963697833728, 22.934057107417441576685929625532, 24.49454979522760405663892506249, 25.271469793929650092873637542, 26.06357209030074734796113752826, 27.17301526627263270510093532279

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