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Properties

Label 1-304-304.131-r1-0-0
Degree $1$
Conductor $304$
Sign $0.523 - 0.851i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.523 - 0.851i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ 0.523 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.201070541 - 1.230277800i\)
\(L(\frac12)\) \(\approx\) \(2.201070541 - 1.230277800i\)
\(L(1)\) \(\approx\) \(1.402763495 - 0.3089415132i\)
\(L(1)\) \(\approx\) \(1.402763495 - 0.3089415132i\)

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

−25.54518686727985553359308300244, −24.42218525021306902883772308484, −23.352656545471543210132668251977, −22.60920218876614131010204582892, −21.70858995608932802123630381146, −20.37170899600972828772277414787, −19.9673263293717182862946983216, −19.10199012033016486254866558752, −18.251959865432545768354042471537, −16.97677865762770307671074115589, −15.86800966378705111514687688426, −15.15657585433466562939979400328, −14.26035294263238606027350255729, −13.46591792754299119657710312451, −12.440629075002440415154662552656, −10.99878369992690651226369855733, −10.37274747761682511521217608134, −9.17470969013555276009111122356, −8.26970918391712853205180473236, −7.07757721518517369689395121232, −6.57901393750785896855623594096, −4.42940025761829615287257386691, −3.73276660358794580368952466723, −2.80712977128674868362130724347, −1.22899172561641568122324279131, 0.76487185402217474297769360486, 2.14330106264064640572472380114, 3.46609112554339721386526610251, 4.24750846526702676478779046284, 5.78555492993724194549573268867, 6.93447761099969130852349978219, 8.155469837137481068247082998762, 8.98632430977673163356579218241, 9.37739290780403853888483513542, 11.18505805364721685789073331460, 12.06035456455162108467391544, 13.08872661204050269730641846613, 13.72812857857730250157947021506, 15.08769679494350649128478037092, 15.66225435476775445277726485803, 16.50796318726989189998508559643, 17.84560421652655300555287417865, 19.16330745855856867319500261594, 19.26275842305220075663469388337, 20.45693808841586463639038220237, 21.132760285095114880786670590228, 22.20955348383923165404033804485, 23.26940951830976670367684518836, 24.45577461301445505244505593140, 24.7807431111577203783786943887

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