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Properties

Label 1-344-344.227-r0-0-0
Degree $1$
Conductor $344$
Sign $-0.958 - 0.284i$
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  + (−0.365 + 0.930i)3-s + (0.0747 + 0.997i)5-s + (−0.5 + 0.866i)7-s + (−0.733 − 0.680i)9-s + (−0.222 + 0.974i)11-s + (−0.826 + 0.563i)13-s + (−0.955 − 0.294i)15-s + (0.0747 − 0.997i)17-s + (0.733 − 0.680i)19-s + (−0.623 − 0.781i)21-s + (−0.955 + 0.294i)23-s + (−0.988 + 0.149i)25-s + (0.900 − 0.433i)27-s + (0.365 + 0.930i)29-s + (0.988 + 0.149i)31-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)3-s + (0.0747 + 0.997i)5-s + (−0.5 + 0.866i)7-s + (−0.733 − 0.680i)9-s + (−0.222 + 0.974i)11-s + (−0.826 + 0.563i)13-s + (−0.955 − 0.294i)15-s + (0.0747 − 0.997i)17-s + (0.733 − 0.680i)19-s + (−0.623 − 0.781i)21-s + (−0.955 + 0.294i)23-s + (−0.988 + 0.149i)25-s + (0.900 − 0.433i)27-s + (0.365 + 0.930i)29-s + (0.988 + 0.149i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $-0.958 - 0.284i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ -0.958 - 0.284i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09165409027 + 0.6309621028i\)
\(L(\frac12)\) \(\approx\) \(-0.09165409027 + 0.6309621028i\)
\(L(1)\) \(\approx\) \(0.5678317490 + 0.4881561646i\)
\(L(1)\) \(\approx\) \(0.5678317490 + 0.4881561646i\)

Euler product

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

−24.393243821059734471385581323379, −23.68089004083584499322918553670, −22.83655731464553764873621386114, −21.88301430555659071057769727433, −20.703575276399707204179636095036, −19.788861002975650006738771853189, −19.26679769201438913715082518258, −18.10216008119879905636840077585, −17.02077580506810133165915478367, −16.74628113599453864881045579760, −15.63403294792477147350933206450, −14.0167937622082673790143432456, −13.48983179528741374542580091035, −12.55443031813955832311159729324, −11.91978011465909673644401677360, −10.62140700673051987623051296492, −9.72235903322945933853048430705, −8.18661009718603298159698854226, −7.83906251230127247704502417475, −6.38274193250979897970483578273, −5.66694788931972318681399005393, −4.44386494590008896254592464529, −3.0553208178686711710626037138, −1.52686043326367584378159943415, −0.41001799298837500234504419599, 2.367885692174217576753813289796, 3.15421369531160257706060674648, 4.527918598043135955315676444, 5.464991851180899962790374825, 6.5652024759559131169929941499, 7.477898945661736488582251674872, 9.192366003554238376596310128600, 9.68858139620448612231295656872, 10.59199551683192229881000701410, 11.731844207186333856742479100195, 12.27661056828507140386931800576, 13.93285947370661208840005398015, 14.6841314393085913352165087459, 15.64148418034342529202671931874, 16.07178460695847812744355882906, 17.55161848477556855834678883511, 18.03094867366848252745609302039, 19.17018723385390505202797222769, 20.09615786099685285650704216138, 21.204291179536714225353990667038, 22.04090722267342997817404319284, 22.48275172725174662283313770149, 23.267961116997497017718408775962, 24.55889810979140622596519109656, 25.715647585386237773808933474765

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