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Properties

Label 1-344-344.75-r0-0-0
Degree $1$
Conductor $344$
Sign $0.526 + 0.849i$
Analytic cond. $1.59752$
Root an. cond. $1.59752$
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  + (0.222 + 0.974i)3-s + (0.623 − 0.781i)5-s + 7-s + (−0.900 + 0.433i)9-s + (−0.900 + 0.433i)11-s + (−0.623 + 0.781i)13-s + (0.900 + 0.433i)15-s + (0.623 + 0.781i)17-s + (0.900 + 0.433i)19-s + (0.222 + 0.974i)21-s + (0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.623 − 0.781i)27-s + (−0.222 + 0.974i)29-s + (0.222 − 0.974i)31-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)3-s + (0.623 − 0.781i)5-s + 7-s + (−0.900 + 0.433i)9-s + (−0.900 + 0.433i)11-s + (−0.623 + 0.781i)13-s + (0.900 + 0.433i)15-s + (0.623 + 0.781i)17-s + (0.900 + 0.433i)19-s + (0.222 + 0.974i)21-s + (0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.623 − 0.781i)27-s + (−0.222 + 0.974i)29-s + (0.222 − 0.974i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.526 + 0.849i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ 0.526 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.379857575 + 0.7680503410i\)
\(L(\frac12)\) \(\approx\) \(1.379857575 + 0.7680503410i\)
\(L(1)\) \(\approx\) \(1.240676131 + 0.3762137605i\)
\(L(1)\) \(\approx\) \(1.240676131 + 0.3762137605i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 + (0.900 + 0.433i)T \)
23 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (0.222 - 0.974i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (-0.222 - 0.974i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−24.93093847589939013820657696851, −23.90984788364054262544941898647, −23.12440167448282706626124382101, −22.167520340656975946802493109050, −21.09989721703870717637166558272, −20.40083069970928074730519819180, −19.18968664167860558450697018470, −18.40845197040937871607230183628, −17.81026583393120353016019580559, −17.10932447557715611985931803864, −15.526023770979663285566522939746, −14.608890906332347526580824555784, −13.84720495669689138990890280032, −13.174441345961939756521704879856, −11.93024960670571337172237727370, −11.11571786485659013472808647663, −10.09582113016966445066224800788, −8.86778327030003992593995298531, −7.60874184008545190034626070351, −7.31010551977447525076658923746, −5.79816261234930476396279492680, −5.151412084212488067199914962833, −3.07187971124427852869354211753, −2.46585036315805347800596224115, −1.07768851450044533031310818967, 1.56023035157123405932158701090, 2.73889955043245524845982967093, 4.31915864413109384534503458296, 4.980507790856618625206422335802, 5.78143401685643193776615599350, 7.58137561704164845511275106229, 8.44820387810230812321126991716, 9.432701597074763830051993240750, 10.1655185968804438109476113703, 11.18368440141852399281329102031, 12.25891615192406939524698434920, 13.36225303490177059518086293858, 14.42734654162115315836001692759, 14.96664318394439250573617683466, 16.23331242785099900181146601748, 16.85887810007572726778502326994, 17.68933712357298194269215468609, 18.787530380119689179484820019404, 20.15111761449961550884225652761, 20.72817556313726706482343111107, 21.35521015571720984951072060584, 22.06233428617294252821016116168, 23.36542898488803618535488028959, 24.18473414776751939597167372840, 25.09346029296013481955002105090

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