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Properties

Label 2-304-76.75-c1-0-5
Degree $2$
Conductor $304$
Sign $0.991 - 0.128i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.64·3-s + 1.73i·7-s + 4.00·9-s − 3.46i·11-s + 4.58i·13-s + 3·17-s + (−2.64 − 3.46i)19-s + 4.58i·21-s − 5.19i·23-s − 5·25-s + 2.64·27-s + 4.58i·29-s − 5.29·31-s − 9.16i·33-s − 9.16i·37-s + ⋯
L(s)  = 1  + 1.52·3-s + 0.654i·7-s + 1.33·9-s − 1.04i·11-s + 1.27i·13-s + 0.727·17-s + (−0.606 − 0.794i)19-s + 0.999i·21-s − 1.08i·23-s − 25-s + 0.509·27-s + 0.850i·29-s − 0.950·31-s − 1.59i·33-s − 1.50i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.991 - 0.128i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02210 + 0.130254i\)
\(L(\frac12)\) \(\approx\) \(2.02210 + 0.130254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (2.64 + 3.46i)T \)
good3 \( 1 - 2.64T + 3T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 4.58iT - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
23 \( 1 + 5.19iT - 23T^{2} \)
29 \( 1 - 4.58iT - 29T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 + 9.16iT - 37T^{2} \)
41 \( 1 - 9.16iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 4.58iT - 53T^{2} \)
59 \( 1 - 7.93T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 2.64T + 67T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 + 6.92iT - 83T^{2} \)
89 \( 1 - 18.3iT - 89T^{2} \)
97 \( 1 - 9.16iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.78146688431203205303171302886, −10.76485744450522398152186168439, −9.431289442079309473702611634864, −8.908580702813959829632085004468, −8.197166136460573239315518007286, −7.08497936416952254733109559369, −5.84837517283389699269946374500, −4.27670906700932501796249552188, −3.14210296018590805661309531729, −2.04937680750422649026388617953, 1.85587376102068615898652805644, 3.26302997517606004399166295917, 4.14887493503137114582602114231, 5.70268222407413260784561238337, 7.41742151558505521450733070441, 7.74057098591661363622282987027, 8.791420101539247436170082518805, 9.936836910631408210227285120107, 10.27190639601475947582000801981, 11.86019397023259963916351964116

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