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Properties

Label 1-344-344.155-r0-0-0
Degree $1$
Conductor $344$
Sign $-0.433 + 0.901i$
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  + (−0.826 + 0.563i)3-s + (−0.733 − 0.680i)5-s + (−0.5 − 0.866i)7-s + (0.365 − 0.930i)9-s + (0.623 + 0.781i)11-s + (−0.955 + 0.294i)13-s + (0.988 + 0.149i)15-s + (−0.733 + 0.680i)17-s + (−0.365 − 0.930i)19-s + (0.900 + 0.433i)21-s + (0.988 − 0.149i)23-s + (0.0747 + 0.997i)25-s + (0.222 + 0.974i)27-s + (0.826 + 0.563i)29-s + (−0.0747 + 0.997i)31-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)3-s + (−0.733 − 0.680i)5-s + (−0.5 − 0.866i)7-s + (0.365 − 0.930i)9-s + (0.623 + 0.781i)11-s + (−0.955 + 0.294i)13-s + (0.988 + 0.149i)15-s + (−0.733 + 0.680i)17-s + (−0.365 − 0.930i)19-s + (0.900 + 0.433i)21-s + (0.988 − 0.149i)23-s + (0.0747 + 0.997i)25-s + (0.222 + 0.974i)27-s + (0.826 + 0.563i)29-s + (−0.0747 + 0.997i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $-0.433 + 0.901i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ -0.433 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1884485128 + 0.2998045278i\)
\(L(\frac12)\) \(\approx\) \(0.1884485128 + 0.2998045278i\)
\(L(1)\) \(\approx\) \(0.5682401407 + 0.07423268547i\)
\(L(1)\) \(\approx\) \(0.5682401407 + 0.07423268547i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (-0.826 + 0.563i)T \)
5 \( 1 + (-0.733 - 0.680i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (-0.955 + 0.294i)T \)
17 \( 1 + (-0.733 + 0.680i)T \)
19 \( 1 + (-0.365 - 0.930i)T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (0.826 + 0.563i)T \)
31 \( 1 + (-0.0747 + 0.997i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (0.0747 + 0.997i)T \)
67 \( 1 + (0.365 + 0.930i)T \)
71 \( 1 + (-0.988 - 0.149i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.826 - 0.563i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−24.71592520677363460890156370726, −23.63982674317607251457907242514, −22.635148866654019496754340861350, −22.34095032121742995337643737852, −21.399684425117761518643992388437, −19.7877924486280730537833556086, −19.06987305702590783023204281795, −18.59058408817172718694219010837, −17.50738214327755861493866730754, −16.56848126748729577273933379143, −15.68467187095343117885786378014, −14.80036853762864698231565067317, −13.64564946294164503006371833816, −12.49729221780667301844823490523, −11.83553257354663170060140736860, −11.1171816353087856277676546837, −10.03298694500352257521963376917, −8.7342124848339667675327059425, −7.605927084525826942250353393362, −6.684362687094832341223635148093, −5.91702236360116232864998902318, −4.72569666728455543657142415059, −3.296585099827032883147276850809, −2.18239025139232666686622937411, −0.26685720815826150348927522119, 1.25989252533101392960045189965, 3.332313209120141050972014398760, 4.559231716382709846931830429433, 4.78477677838932134518197693975, 6.62065971998699401128320754859, 7.09520372143684352802798307858, 8.67624180909920566480134217324, 9.58960012492434737847897156974, 10.53761114092179317645661925680, 11.477171134777197796314441764527, 12.395425788534715647867329015, 13.06865294854000095971660742194, 14.61494154506185234999396547291, 15.451923643575129706869752964139, 16.30457785560308606039167722014, 17.17617211974448768997700722249, 17.531324748529772044656984485874, 19.26647623804875522936383146821, 19.85524281708104517382510366903, 20.68723478417718832546221030010, 21.84104349160721793789442817439, 22.53836205353627022735867707231, 23.51262359476237983705653819837, 23.89706190166179119463621090309, 25.09747319509542682851470466152

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