Properties

Label 1-373-373.52-r0-0-0
Degree $1$
Conductor $373$
Sign $0.754 - 0.655i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (0.0168 + 0.999i)2-s + (0.638 − 0.769i)3-s + (−0.999 + 0.0337i)4-s + (0.283 − 0.959i)5-s + (0.780 + 0.625i)6-s + (0.979 − 0.201i)7-s + (−0.0506 − 0.998i)8-s + (−0.184 − 0.982i)9-s + (0.963 + 0.266i)10-s + (−0.839 − 0.543i)11-s + (−0.612 + 0.790i)12-s + (−0.0506 + 0.998i)13-s + (0.217 + 0.975i)14-s + (−0.557 − 0.830i)15-s + (0.997 − 0.0675i)16-s + (0.528 + 0.848i)17-s + ⋯
L(s)  = 1  + (0.0168 + 0.999i)2-s + (0.638 − 0.769i)3-s + (−0.999 + 0.0337i)4-s + (0.283 − 0.959i)5-s + (0.780 + 0.625i)6-s + (0.979 − 0.201i)7-s + (−0.0506 − 0.998i)8-s + (−0.184 − 0.982i)9-s + (0.963 + 0.266i)10-s + (−0.839 − 0.543i)11-s + (−0.612 + 0.790i)12-s + (−0.0506 + 0.998i)13-s + (0.217 + 0.975i)14-s + (−0.557 − 0.830i)15-s + (0.997 − 0.0675i)16-s + (0.528 + 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.754 - 0.655i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.754 - 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421098884 - 0.5310565324i\)
\(L(\frac12)\) \(\approx\) \(1.421098884 - 0.5310565324i\)
\(L(1)\) \(\approx\) \(1.259178042 - 0.07953518494i\)
\(L(1)\) \(\approx\) \(1.259178042 - 0.07953518494i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.0168 + 0.999i)T \)
3 \( 1 + (0.638 - 0.769i)T \)
5 \( 1 + (0.283 - 0.959i)T \)
7 \( 1 + (0.979 - 0.201i)T \)
11 \( 1 + (-0.839 - 0.543i)T \)
13 \( 1 + (-0.0506 + 0.998i)T \)
17 \( 1 + (0.528 + 0.848i)T \)
19 \( 1 + (-0.440 - 0.897i)T \)
23 \( 1 + (0.347 - 0.937i)T \)
29 \( 1 + (0.0843 + 0.996i)T \)
31 \( 1 + (-0.612 - 0.790i)T \)
37 \( 1 + (0.409 - 0.912i)T \)
41 \( 1 + (0.151 - 0.988i)T \)
43 \( 1 + (-0.999 + 0.0337i)T \)
47 \( 1 + (0.736 + 0.676i)T \)
53 \( 1 + (-0.184 + 0.982i)T \)
59 \( 1 + (-0.905 - 0.425i)T \)
61 \( 1 + (0.638 - 0.769i)T \)
67 \( 1 + (0.688 + 0.724i)T \)
71 \( 1 + (0.470 + 0.882i)T \)
73 \( 1 + (0.943 - 0.331i)T \)
79 \( 1 + (0.963 + 0.266i)T \)
83 \( 1 + (-0.184 + 0.982i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.528 + 0.848i)T \)
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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

−25.08828087300528621640822709055, −23.3831828175673037415339590006, −22.781831321329590710803211691877, −21.78302852922426630759099008613, −21.1267672507599536575083687276, −20.56880638316811531091697867269, −19.63222762937777857297707304763, −18.5230635377721230181288734540, −18.048991530102671538644838951555, −16.97980955808027807517782467965, −15.36327981221526098492400862649, −14.84361592868787518932007053011, −13.98950779901840396515110021965, −13.177907003976868140613098262537, −11.83697812081978543731890552131, −10.91315535206767515100254766097, −10.21070258806977779269987323630, −9.589725938899667370946121260233, −8.21848575313999419107261618201, −7.64269642851578883167174325962, −5.538970420349165792280691516041, −4.85141378259319587074502150170, −3.53323486275838622254635890507, −2.73288973100488039548418593360, −1.81982119743230565171238733873, 0.8912058424070358774724098564, 2.14872277990955881260352573360, 3.916693810466954755835244723988, 4.92645998847617967763206610968, 5.91525661804189163178397349604, 7.0496139749481362457666157741, 8.02386805408958039939902379824, 8.6116596818806216595149999559, 9.35221808217869792846832235030, 10.88292482643810543677492391736, 12.38454679667628870723374454082, 13.02217899842226144000177184206, 13.93100917991603258318380802282, 14.52290502254886657251375629351, 15.56461997810154898360263375757, 16.7139965506540698001380020165, 17.29659849945080930197689588965, 18.296479623493454894283073835440, 18.96337131763619707270284199209, 20.11960724621979707616324491514, 21.12258653215973720619931711035, 21.70878407837940392088644160784, 23.54893592763016956501648511446, 23.807999844997373073005893438190, 24.37652544992640320181702682611

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