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Properties

Label 1-1200-1200.323-r1-0-0
Degree $1$
Conductor $1200$
Sign $-0.495 + 0.868i$
Analytic cond. $128.957$
Root an. cond. $128.957$
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  − i·7-s + (−0.951 − 0.309i)11-s + (−0.309 − 0.951i)13-s + (−0.587 − 0.809i)17-s + (−0.587 − 0.809i)19-s + (−0.951 − 0.309i)23-s + (0.587 − 0.809i)29-s + (0.809 − 0.587i)31-s + (−0.309 − 0.951i)37-s + (0.309 + 0.951i)41-s − 43-s + (−0.587 + 0.809i)47-s − 49-s + (0.809 + 0.587i)53-s + (−0.951 + 0.309i)59-s + ⋯
L(s)  = 1  − i·7-s + (−0.951 − 0.309i)11-s + (−0.309 − 0.951i)13-s + (−0.587 − 0.809i)17-s + (−0.587 − 0.809i)19-s + (−0.951 − 0.309i)23-s + (0.587 − 0.809i)29-s + (0.809 − 0.587i)31-s + (−0.309 − 0.951i)37-s + (0.309 + 0.951i)41-s − 43-s + (−0.587 + 0.809i)47-s − 49-s + (0.809 + 0.587i)53-s + (−0.951 + 0.309i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.495 + 0.868i$
Analytic conductor: \(128.957\)
Root analytic conductor: \(128.957\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1200,\ (1:\ ),\ -0.495 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2118222085 - 0.3646786383i\)
\(L(\frac12)\) \(\approx\) \(-0.2118222085 - 0.3646786383i\)
\(L(1)\) \(\approx\) \(0.7624589519 - 0.2945160995i\)
\(L(1)\) \(\approx\) \(0.7624589519 - 0.2945160995i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - iT \)
11 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.587 - 0.809i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + (0.587 - 0.809i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (0.951 + 0.309i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−21.545300493216451470833095667439, −20.910897065134628805211345831949, −19.88833260015109005988434875883, −19.14024676372547697089735212756, −18.456954751324924022543203451275, −17.78817607758047276986749788446, −16.89321952569601105539063679273, −15.977656716489471201348556307556, −15.39532771897350019408756960063, −14.610765368266759710077843945983, −13.78198953355170591820612679323, −12.78660260884330689135857283165, −12.20080247440400024965276756970, −11.44564649607500344143930510759, −10.35792486926740392179788720536, −9.7970254886586253150187350769, −8.56333565465992627063125538042, −8.29718650260824976688836876605, −6.997405216579594497918045087492, −6.24541037087501831326549203489, −5.31798272908267843924827343377, −4.523925066283257441604699094739, −3.43949021337995201819678511129, −2.28815269695455853124353830801, −1.72175747475929907218211395278, 0.1061766328648679085836901015, 0.750017534228100302575223305008, 2.303696328990803848172628631323, 3.03017728800683296126016163092, 4.24688366388441788431185813577, 4.89478770812828102931227303141, 5.97287315501923143961656298462, 6.87405907805182753518280128461, 7.77640589910853327613934807026, 8.316442584587693087522145509674, 9.55875394948342552868098288114, 10.29821380431514850883924038306, 10.915688688043087339781821843591, 11.791095129439522841622836566785, 12.91720599318472418844884360343, 13.40869408173868293709901356601, 14.13529435417104863540001613044, 15.19139568767215070258539857279, 15.837891789291931952263675057855, 16.59896662739036830971107342079, 17.59677821239767216955708371100, 17.943959916531285041266211959420, 19.024759049018727220792351419438, 19.86558757666002097908247038155, 20.372780279549695913586412374723

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