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Properties

Label 2-304-19.17-c1-0-1
Degree $2$
Conductor $304$
Sign $-0.466 - 0.884i$
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  + (0.986 + 0.827i)3-s + (−3.98 + 1.45i)5-s + (1.68 + 2.92i)7-s + (−0.233 − 1.32i)9-s + (−0.247 + 0.429i)11-s + (−2.70 + 2.27i)13-s + (−5.13 − 1.86i)15-s + (−0.689 + 3.90i)17-s + (−2.26 + 3.72i)19-s + (−0.753 + 4.27i)21-s + (0.886 + 0.322i)23-s + (9.96 − 8.35i)25-s + (2.79 − 4.84i)27-s + (−0.463 − 2.62i)29-s + (3.41 + 5.91i)31-s + ⋯
L(s)  = 1  + (0.569 + 0.477i)3-s + (−1.78 + 0.649i)5-s + (0.637 + 1.10i)7-s + (−0.0777 − 0.440i)9-s + (−0.0747 + 0.129i)11-s + (−0.750 + 0.629i)13-s + (−1.32 − 0.482i)15-s + (−0.167 + 0.947i)17-s + (−0.519 + 0.854i)19-s + (−0.164 + 0.932i)21-s + (0.184 + 0.0672i)23-s + (1.99 − 1.67i)25-s + (0.537 − 0.931i)27-s + (−0.0860 − 0.488i)29-s + (0.613 + 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.466 - 0.884i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.466 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.521209 + 0.863717i\)
\(L(\frac12)\) \(\approx\) \(0.521209 + 0.863717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.26 - 3.72i)T \)
good3 \( 1 + (-0.986 - 0.827i)T + (0.520 + 2.95i)T^{2} \)
5 \( 1 + (3.98 - 1.45i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-1.68 - 2.92i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.247 - 0.429i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.70 - 2.27i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.689 - 3.90i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-0.886 - 0.322i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.463 + 2.62i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.41 - 5.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.73T + 37T^{2} \)
41 \( 1 + (1.85 + 1.55i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-11.2 + 4.10i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.194 + 1.10i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (0.951 + 0.346i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.14 - 6.49i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (8.89 + 3.23i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.84 - 10.4i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-8.85 + 3.22i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-2.49 - 2.09i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (4.41 + 3.70i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (3.89 + 6.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.05 + 1.72i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.248 + 1.41i)T + (-91.1 - 33.1i)T^{2} \)
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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

−12.04689836286689603246214300771, −11.22202227117255847199774623632, −10.22667006605928933717628999996, −8.926702847112744555758806096262, −8.320813289500003514288334889151, −7.44841642459841845306538660427, −6.22053475922379895707417197262, −4.56929452995958590826226378119, −3.77041763939517322957857785322, −2.55053085749554923165369374447, 0.70877849529546939147926911441, 2.84894829855023712950767745967, 4.29752252541440130895850078058, 4.91652485007182728952297338247, 7.08532695534656093136423677832, 7.74508924915231132433661316013, 8.135980796944113516393226921854, 9.296853106741279059957978670880, 10.88078897867253785824427413024, 11.28722901851688889735117351312

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