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Properties

Label 1-304-304.181-r1-0-0
Degree $1$
Conductor $304$
Sign $-0.999 + 0.0204i$
Analytic cond. $32.6693$
Root an. cond. $32.6693$
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  + (0.984 + 0.173i)3-s + (−0.642 + 0.766i)5-s + (0.5 + 0.866i)7-s + (0.939 + 0.342i)9-s + (−0.866 − 0.5i)11-s + (−0.984 + 0.173i)13-s + (−0.766 + 0.642i)15-s + (−0.939 + 0.342i)17-s + (0.342 + 0.939i)21-s + (−0.766 + 0.642i)23-s + (−0.173 − 0.984i)25-s + (0.866 + 0.5i)27-s + (0.342 − 0.939i)29-s + (0.5 + 0.866i)31-s + (−0.766 − 0.642i)33-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)3-s + (−0.642 + 0.766i)5-s + (0.5 + 0.866i)7-s + (0.939 + 0.342i)9-s + (−0.866 − 0.5i)11-s + (−0.984 + 0.173i)13-s + (−0.766 + 0.642i)15-s + (−0.939 + 0.342i)17-s + (0.342 + 0.939i)21-s + (−0.766 + 0.642i)23-s + (−0.173 − 0.984i)25-s + (0.866 + 0.5i)27-s + (0.342 − 0.939i)29-s + (0.5 + 0.866i)31-s + (−0.766 − 0.642i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.999 + 0.0204i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ -0.999 + 0.0204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01015660962 + 0.9925077588i\)
\(L(\frac12)\) \(\approx\) \(0.01015660962 + 0.9925077588i\)
\(L(1)\) \(\approx\) \(0.9881369057 + 0.4059814653i\)
\(L(1)\) \(\approx\) \(0.9881369057 + 0.4059814653i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.984 + 0.173i)T \)
5 \( 1 + (-0.642 + 0.766i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.342 - 0.939i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (-0.642 - 0.766i)T \)
59 \( 1 + (0.342 + 0.939i)T \)
61 \( 1 + (-0.642 - 0.766i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−24.394908278265911305642299291247, −24.09215942986422514095382107557, −23.10827835168831148460962294430, −21.813674880494077681792232694015, −20.55131090271153638027546362560, −20.31813925370291795072053136576, −19.542562878848589866638514543622, −18.388886612615500858797694025459, −17.43527553711325369450604724348, −16.32133621547269771928476429602, −15.40143199275234057055808871641, −14.59868986627405281278561197728, −13.507859874731681506728990587702, −12.80988186142340672859907550682, −11.80563479777226050908529962311, −10.460897797155432128202181420319, −9.54394262731025752114078886100, −8.32018371578326264528944880004, −7.76826228970636469514862096056, −6.86592732666909602192995212701, −4.82878218412357081097317626532, −4.35357148555448699195044367656, −2.93071574010006942041428418150, −1.68530416165344476665095186954, −0.23769663878315513238403924633, 2.109082457859900166395964813577, 2.80970095731416848942392179201, 4.01657334108837230353249973925, 5.160009821519519111151606409385, 6.65435010453229304612917203708, 7.827781515594117799039839423063, 8.35152483035104820017344293276, 9.553148319591951552735110559668, 10.58432836946067905508833124084, 11.58502100798756764004683755567, 12.63378264222108687222585102370, 13.83857344939343565202221905804, 14.62220360126103760454832729319, 15.46939972843097533266809911456, 15.93150216274376696480822074461, 17.66595447608384476283593966957, 18.485364550399766701536093038118, 19.34095127583582890154990437174, 19.90399685158609638094261011662, 21.32943674613858992375783188198, 21.63897524887858963645322345694, 22.771754144525424083336205458471, 24.071271532848967768450883274010, 24.54164210907011081688749416310, 25.68925354583432162826286717680

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