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Properties

Label 2-304-16.13-c1-0-25
Degree $2$
Conductor $304$
Sign $-0.972 + 0.231i$
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  + (−0.738 − 1.20i)2-s + (−0.325 − 0.325i)3-s + (−0.909 + 1.78i)4-s + (−0.432 + 0.432i)5-s + (−0.152 + 0.632i)6-s − 0.877i·7-s + (2.82 − 0.218i)8-s − 2.78i·9-s + (0.840 + 0.202i)10-s + (−0.148 + 0.148i)11-s + (0.875 − 0.283i)12-s + (−4.57 − 4.57i)13-s + (−1.05 + 0.647i)14-s + 0.281·15-s + (−2.34 − 3.24i)16-s − 4.80·17-s + ⋯
L(s)  = 1  + (−0.522 − 0.852i)2-s + (−0.187 − 0.187i)3-s + (−0.454 + 0.890i)4-s + (−0.193 + 0.193i)5-s + (−0.0621 + 0.258i)6-s − 0.331i·7-s + (0.997 − 0.0771i)8-s − 0.929i·9-s + (0.265 + 0.0639i)10-s + (−0.0447 + 0.0447i)11-s + (0.252 − 0.0818i)12-s + (−1.26 − 1.26i)13-s + (−0.282 + 0.173i)14-s + 0.0725·15-s + (−0.586 − 0.810i)16-s − 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.972 + 0.231i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.972 + 0.231i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0637758 - 0.542975i\)
\(L(\frac12)\) \(\approx\) \(0.0637758 - 0.542975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.738 + 1.20i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.325 + 0.325i)T + 3iT^{2} \)
5 \( 1 + (0.432 - 0.432i)T - 5iT^{2} \)
7 \( 1 + 0.877iT - 7T^{2} \)
11 \( 1 + (0.148 - 0.148i)T - 11iT^{2} \)
13 \( 1 + (4.57 + 4.57i)T + 13iT^{2} \)
17 \( 1 + 4.80T + 17T^{2} \)
23 \( 1 + 5.90iT - 23T^{2} \)
29 \( 1 + (4.03 + 4.03i)T + 29iT^{2} \)
31 \( 1 - 9.44T + 31T^{2} \)
37 \( 1 + (2.05 - 2.05i)T - 37iT^{2} \)
41 \( 1 + 11.4iT - 41T^{2} \)
43 \( 1 + (3.98 - 3.98i)T - 43iT^{2} \)
47 \( 1 + 6.46T + 47T^{2} \)
53 \( 1 + (-1.81 + 1.81i)T - 53iT^{2} \)
59 \( 1 + (-3.14 + 3.14i)T - 59iT^{2} \)
61 \( 1 + (4.49 + 4.49i)T + 61iT^{2} \)
67 \( 1 + (-10.7 - 10.7i)T + 67iT^{2} \)
71 \( 1 + 2.42iT - 71T^{2} \)
73 \( 1 + 3.14iT - 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + (-4.96 - 4.96i)T + 83iT^{2} \)
89 \( 1 - 6.03iT - 89T^{2} \)
97 \( 1 - 6.45T + 97T^{2} \)
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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

−11.27875773106404155164769276387, −10.34008780260774975857697165059, −9.636589780050628113260036547559, −8.543340051991907589872709058378, −7.55991866667287175353354216371, −6.62398608878006917163612076571, −4.96921987023113968249806493722, −3.70994438493460075274943472484, −2.47993580474835307167552883376, −0.47134715548434740423028792993, 2.10182293166111115490723752473, 4.47788936051466030613339944327, 5.11552641732505632998006628753, 6.44357172164378356874503976973, 7.35629188903329855069876758938, 8.316558652771655436497818145112, 9.265980982459214331143362522862, 10.03370847466960934068198145038, 11.12894657299627068824592416947, 11.95031605068813807124274629335

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