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Properties

Label 2-304-304.197-c1-0-32
Degree $2$
Conductor $304$
Sign $-0.215 + 0.976i$
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  + (1.40 − 0.159i)2-s + (−2.81 − 0.755i)3-s + (1.94 − 0.448i)4-s + (0.559 − 2.08i)5-s + (−4.08 − 0.612i)6-s + 1.48i·7-s + (2.66 − 0.940i)8-s + (4.78 + 2.76i)9-s + (0.453 − 3.02i)10-s + (−3.96 − 3.96i)11-s + (−5.83 − 0.209i)12-s + (−1.36 − 5.10i)13-s + (0.237 + 2.09i)14-s + (−3.15 + 5.46i)15-s + (3.59 − 1.74i)16-s + (−2.62 − 4.54i)17-s + ⋯
L(s)  = 1  + (0.993 − 0.112i)2-s + (−1.62 − 0.436i)3-s + (0.974 − 0.224i)4-s + (0.250 − 0.934i)5-s + (−1.66 − 0.249i)6-s + 0.562i·7-s + (0.943 − 0.332i)8-s + (1.59 + 0.920i)9-s + (0.143 − 0.956i)10-s + (−1.19 − 1.19i)11-s + (−1.68 − 0.0603i)12-s + (−0.379 − 1.41i)13-s + (0.0634 + 0.558i)14-s + (−0.814 + 1.41i)15-s + (0.899 − 0.436i)16-s + (−0.637 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.850444 - 1.05815i\)
\(L(\frac12)\) \(\approx\) \(0.850444 - 1.05815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.159i)T \)
19 \( 1 + (-1.93 - 3.90i)T \)
good3 \( 1 + (2.81 + 0.755i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.559 + 2.08i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 - 1.48iT - 7T^{2} \)
11 \( 1 + (3.96 + 3.96i)T + 11iT^{2} \)
13 \( 1 + (1.36 + 5.10i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.62 + 4.54i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.994 - 0.573i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.49 - 5.56i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 - 3.13T + 31T^{2} \)
37 \( 1 + (-6.97 - 6.97i)T + 37iT^{2} \)
41 \( 1 + (6.37 - 3.67i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.30 + 4.85i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.61 + 7.98i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.480 - 0.128i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.922 - 3.44i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.367 - 1.37i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.70 - 6.37i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.96 - 2.86i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.65 - 0.956i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.79 + 3.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.60 + 1.60i)T - 83iT^{2} \)
89 \( 1 + (-10.6 - 6.16i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.786 - 1.36i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.76348038559079853586162458401, −10.82438147220115380975276516111, −10.10905471878806470463032970744, −8.374579036189635249126043237639, −7.24180722438491390919511086689, −5.97843535352051683236369335591, −5.34691127171355470067114164933, −4.95526498281272232888577771322, −2.87363484868962390552035159144, −0.899214550027149506928740963266, 2.34044080563126968500474988172, 4.26889399580727755176406585430, 4.79401331604356269077723433154, 6.06905139984285542305171225125, 6.76703821340384677698699827139, 7.47790267637848626528101016976, 9.742766470827742035635853212620, 10.60888039309532769312087825202, 11.01297277232593249133361993649, 11.92522954698566501571862676249

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