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Properties

Label 1-373-373.165-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: $\chi_{373} (165, \cdot )$
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

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

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