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Properties

Label 2-304-19.11-c1-0-6
Degree $2$
Conductor $304$
Sign $0.321 + 0.946i$
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  + (0.5 − 0.866i)3-s + (2 − 3.46i)5-s + (1 + 1.73i)9-s − 3·11-s + (−1 − 1.73i)13-s + (−1.99 − 3.46i)15-s + (−1 + 1.73i)17-s + (−0.5 − 4.33i)19-s + (3 + 5.19i)23-s + (−5.49 − 9.52i)25-s + 5·27-s + (2 + 3.46i)29-s + 10·31-s + (−1.5 + 2.59i)33-s + 2·37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.894 − 1.54i)5-s + (0.333 + 0.577i)9-s − 0.904·11-s + (−0.277 − 0.480i)13-s + (−0.516 − 0.894i)15-s + (−0.242 + 0.420i)17-s + (−0.114 − 0.993i)19-s + (0.625 + 1.08i)23-s + (−1.09 − 1.90i)25-s + 0.962·27-s + (0.371 + 0.643i)29-s + 1.79·31-s + (−0.261 + 0.452i)33-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.321 + 0.946i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.321 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27692 - 0.914482i\)
\(L(\frac12)\) \(\approx\) \(1.27692 - 0.914482i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 4.33i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.5 - 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.5 + 7.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5T + 83T^{2} \)
89 \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.72198242400185260630345459325, −10.39000189893979350036112191182, −9.636470324861114824405229001086, −8.534495941313028896513287988989, −7.972884941498347770618588187395, −6.64716098032331139238775547717, −5.25777917572658162765240638035, −4.76808597674885143024624711630, −2.62006520481166113612200825421, −1.29217626946479508523343033710, 2.34711947397844923830619801392, 3.30921564315505916031443927667, 4.76325260119392571148515262340, 6.23671848197268480345670693083, 6.82249317969547774926746422880, 8.079281494323192069601970622651, 9.412106672404616378564180251946, 10.12687790120778233184382470633, 10.62440029208520472919420562422, 11.76258072278365347564656193090

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