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Properties

Label 1-148-148.103-r0-0-0
Degree $1$
Conductor $148$
Sign $0.849 + 0.527i$
Analytic cond. $0.687309$
Root an. cond. $0.687309$
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  + (−0.5 − 0.866i)3-s + (−0.866 + 0.5i)5-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + 11-s + (−0.866 + 0.5i)13-s + (0.866 + 0.5i)15-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)19-s + (0.5 − 0.866i)21-s − i·23-s + (0.5 − 0.866i)25-s + 27-s + i·29-s − i·31-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.866 + 0.5i)5-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + 11-s + (−0.866 + 0.5i)13-s + (0.866 + 0.5i)15-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)19-s + (0.5 − 0.866i)21-s − i·23-s + (0.5 − 0.866i)25-s + 27-s + i·29-s − i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $0.849 + 0.527i$
Analytic conductor: \(0.687309\)
Root analytic conductor: \(0.687309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 148,\ (0:\ ),\ 0.849 + 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7838061686 + 0.2235726399i\)
\(L(\frac12)\) \(\approx\) \(0.7838061686 + 0.2235726399i\)
\(L(1)\) \(\approx\) \(0.8388522608 + 0.04083762858i\)
\(L(1)\) \(\approx\) \(0.8388522608 + 0.04083762858i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 - iT \)
29 \( 1 + iT \)
31 \( 1 - iT \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 - T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 - iT \)
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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

−27.76919662682985695650062265613, −27.1552305906962765173343986165, −26.632408997229722145582605258070, −24.96872432879917563055594181460, −24.03863261873788609885545987248, −22.946959112685691565582130156263, −22.41035348228216507250086611200, −20.929113628537611607383187425369, −20.324548023821441355112090767258, −19.35762573203031598476668239313, −17.73988075083992393742304124784, −16.81343279942126955013080206773, −16.20143887963681996994929380479, −14.93769238121703517256832962447, −14.13582186104956567823627576226, −12.30008403519623511406272094393, −11.68304770214305689128764944155, −10.506539456636189346523717073340, −9.499826878444028931675280384543, −8.1708836916127575005666433718, −7.02507230831844069853172790011, −5.33720939446730881577097901399, −4.39472107435367312077538594364, −3.43927342930409744806259617622, −0.855028298863351235917891193207, 1.54133658187264309571428953286, 3.06598824081403427139597760075, 4.76681488515940500335472867767, 6.05841930264207839516117635530, 7.22424718042020686679901684459, 8.04850128659261264651969594283, 9.46226145116692456863773289264, 11.24216283956682409652797428921, 11.748885756456574226371978737270, 12.58293458490759326716425037950, 14.182337956905746746069569687407, 14.91508471106630559294173863867, 16.26356347161031162933015796347, 17.35793856370060072030823037852, 18.34869166496864902033138117118, 19.20097430462553366725781487537, 19.88792991299106994580531729847, 21.690918771881552231887062214680, 22.34142435713322109073025087417, 23.44628371925019253098589255048, 24.263387605893918267042364645, 25.03434912847894745075616499001, 26.27375412949334395457835159632, 27.61542143571508350813769120888, 28.00600903464581692411838002861

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