Properties

Label 1-345-345.248-r0-0-0
Degree $1$
Conductor $345$
Sign $0.242 + 0.970i$
Analytic cond. $1.60217$
Root an. cond. $1.60217$
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  + (−0.540 + 0.841i)2-s + (−0.415 − 0.909i)4-s + (0.755 + 0.654i)7-s + (0.989 + 0.142i)8-s + (−0.841 + 0.540i)11-s + (0.755 − 0.654i)13-s + (−0.959 + 0.281i)14-s + (−0.654 + 0.755i)16-s + (0.909 + 0.415i)17-s + (−0.415 − 0.909i)19-s i·22-s + (0.142 + 0.989i)26-s + (0.281 − 0.959i)28-s + (0.415 − 0.909i)29-s + (−0.142 + 0.989i)31-s + (−0.281 − 0.959i)32-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)2-s + (−0.415 − 0.909i)4-s + (0.755 + 0.654i)7-s + (0.989 + 0.142i)8-s + (−0.841 + 0.540i)11-s + (0.755 − 0.654i)13-s + (−0.959 + 0.281i)14-s + (−0.654 + 0.755i)16-s + (0.909 + 0.415i)17-s + (−0.415 − 0.909i)19-s i·22-s + (0.142 + 0.989i)26-s + (0.281 − 0.959i)28-s + (0.415 − 0.909i)29-s + (−0.142 + 0.989i)31-s + (−0.281 − 0.959i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(345\)    =    \(3 \cdot 5 \cdot 23\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(1.60217\)
Root analytic conductor: \(1.60217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{345} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 345,\ (0:\ ),\ 0.242 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7913406673 + 0.6179075038i\)
\(L(\frac12)\) \(\approx\) \(0.7913406673 + 0.6179075038i\)
\(L(1)\) \(\approx\) \(0.7970703003 + 0.3660332078i\)
\(L(1)\) \(\approx\) \(0.7970703003 + 0.3660332078i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.540 + 0.841i)T \)
7 \( 1 + (0.755 + 0.654i)T \)
11 \( 1 + (-0.841 + 0.540i)T \)
13 \( 1 + (0.755 - 0.654i)T \)
17 \( 1 + (0.909 + 0.415i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (0.281 + 0.959i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.989 - 0.142i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.755 + 0.654i)T \)
59 \( 1 + (-0.654 - 0.755i)T \)
61 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (0.540 - 0.841i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (-0.909 + 0.415i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (0.281 + 0.959i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (-0.281 + 0.959i)T \)
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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

−24.78803712683493478441747612629, −23.570021318754456659780999694740, −23.02171984529658048995104725321, −21.64467735908809881113001098701, −20.97710415551016219091356895290, −20.48881444775367913667638651700, −19.268955169315033263256635151496, −18.52344080874063275924800778536, −17.8213331599402095245979027646, −16.67361615850477965482223264005, −16.15754894510978847948739456677, −14.4837426761059767438431321463, −13.72735551015299122684631567150, −12.7730588498433756744169036431, −11.68560490531079418874454827975, −10.87666108613815668069422197077, −10.20211099615184728424661430250, −8.97584084885930883344529123343, −8.06427284229082364750064335787, −7.31567569597285293888485796792, −5.6765730170604800998966146704, −4.373938852366340819452553563563, −3.46146590826366176616816309466, −2.11748763733388249418141019612, −0.93565706174626694606229710677, 1.219619342391327714334659063050, 2.62310904931528566106631224487, 4.46000653119549322113699024415, 5.385148678138373843257926400889, 6.229634390332704726959512814796, 7.59568411575808347316775236901, 8.2028278212015566598110815791, 9.14815932676995363628071821550, 10.300853474517513820456374746563, 11.05794682068106655236854977419, 12.43145359431870980072751493924, 13.47244597642549018657745536806, 14.543488651690628900143763595964, 15.35389367228495021027133160911, 15.9111433733427692534501344050, 17.20987580670805284473939348353, 17.88290526759119342461332784582, 18.56069235500861742817961242551, 19.51211399557167838513310206830, 20.65826593243534177914121025103, 21.477807464670990727568941661979, 22.74536306363022839133350464671, 23.50984420708101028870787572131, 24.22057406910512143910291918397, 25.277699396747564972589772428340

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