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Properties

Label 2-304-304.227-c1-0-28
Degree $2$
Conductor $304$
Sign $0.997 + 0.0694i$
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  + (1.28 + 0.582i)2-s + (−0.623 − 0.623i)3-s + (1.32 + 1.50i)4-s + (2.23 − 2.23i)5-s + (−0.439 − 1.16i)6-s − 0.609·7-s + (0.825 + 2.70i)8-s − 2.22i·9-s + (4.18 − 1.57i)10-s + (−0.224 − 0.224i)11-s + (0.113 − 1.76i)12-s + (−0.855 + 0.855i)13-s + (−0.784 − 0.355i)14-s − 2.78·15-s + (−0.513 + 3.96i)16-s + 2.35·17-s + ⋯
L(s)  = 1  + (0.911 + 0.412i)2-s + (−0.360 − 0.360i)3-s + (0.660 + 0.751i)4-s + (0.999 − 0.999i)5-s + (−0.179 − 0.476i)6-s − 0.230·7-s + (0.291 + 0.956i)8-s − 0.740i·9-s + (1.32 − 0.498i)10-s + (−0.0677 − 0.0677i)11-s + (0.0327 − 0.508i)12-s + (−0.237 + 0.237i)13-s + (−0.209 − 0.0949i)14-s − 0.719·15-s + (−0.128 + 0.991i)16-s + 0.571·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.997 + 0.0694i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.997 + 0.0694i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17720 - 0.0756845i\)
\(L(\frac12)\) \(\approx\) \(2.17720 - 0.0756845i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.582i)T \)
19 \( 1 + (1.42 - 4.11i)T \)
good3 \( 1 + (0.623 + 0.623i)T + 3iT^{2} \)
5 \( 1 + (-2.23 + 2.23i)T - 5iT^{2} \)
7 \( 1 + 0.609T + 7T^{2} \)
11 \( 1 + (0.224 + 0.224i)T + 11iT^{2} \)
13 \( 1 + (0.855 - 0.855i)T - 13iT^{2} \)
17 \( 1 - 2.35T + 17T^{2} \)
23 \( 1 - 2.70T + 23T^{2} \)
29 \( 1 + (2.24 - 2.24i)T - 29iT^{2} \)
31 \( 1 + 3.24T + 31T^{2} \)
37 \( 1 + (0.590 + 0.590i)T + 37iT^{2} \)
41 \( 1 - 2.72T + 41T^{2} \)
43 \( 1 + (2.03 + 2.03i)T + 43iT^{2} \)
47 \( 1 - 6.19iT - 47T^{2} \)
53 \( 1 + (7.68 + 7.68i)T + 53iT^{2} \)
59 \( 1 + (3.91 - 3.91i)T - 59iT^{2} \)
61 \( 1 + (-1.55 - 1.55i)T + 61iT^{2} \)
67 \( 1 + (5.79 + 5.79i)T + 67iT^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 - 9.08iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + (5.08 - 5.08i)T - 83iT^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 8.89iT - 97T^{2} \)
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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

−12.21434216164100104795435270568, −11.04805003098080656834211004182, −9.721564794827255620280356377609, −8.855664700955289008327458775577, −7.63075897247939122408809192512, −6.45443875707570024648492287958, −5.76902121473554446288763581703, −4.87551241825799300412968326305, −3.45951054962352179341254999689, −1.68165867894105631693167811843, 2.13683165470402838105847050911, 3.17526222143362769654140494637, 4.72514733810180369892742269321, 5.63724055639542443806614145916, 6.50752126524708792013773724452, 7.52183799970001131040042473740, 9.405268861566367651938348159436, 10.22435812719231610232259252116, 10.80048645681448042551585106566, 11.55104064876848737836029694314

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