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Properties

Label 2-304-304.27-c1-0-37
Degree $2$
Conductor $304$
Sign $0.507 - 0.861i$
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  + (−0.542 − 1.30i)2-s + (−0.844 − 3.15i)3-s + (−1.41 + 1.41i)4-s + (−3.53 + 0.948i)5-s + (−3.65 + 2.81i)6-s + 2.14·7-s + (2.61 + 1.07i)8-s + (−6.61 + 3.81i)9-s + (3.15 + 4.10i)10-s + (−0.139 + 0.139i)11-s + (5.65 + 3.25i)12-s + (1.17 − 4.38i)13-s + (−1.16 − 2.79i)14-s + (5.97 + 10.3i)15-s + (−0.0150 − 3.99i)16-s + (−2.66 + 4.61i)17-s + ⋯
L(s)  = 1  + (−0.383 − 0.923i)2-s + (−0.487 − 1.81i)3-s + (−0.705 + 0.708i)4-s + (−1.58 + 0.423i)5-s + (−1.49 + 1.14i)6-s + 0.810·7-s + (0.924 + 0.380i)8-s + (−2.20 + 1.27i)9-s + (0.998 + 1.29i)10-s + (−0.0421 + 0.0421i)11-s + (1.63 + 0.938i)12-s + (0.325 − 1.21i)13-s + (−0.310 − 0.748i)14-s + (1.54 + 2.67i)15-s + (−0.00377 − 0.999i)16-s + (−0.646 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.507 - 0.861i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.507 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0543699 + 0.0310632i\)
\(L(\frac12)\) \(\approx\) \(0.0543699 + 0.0310632i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.542 + 1.30i)T \)
19 \( 1 + (4.18 - 1.21i)T \)
good3 \( 1 + (0.844 + 3.15i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (3.53 - 0.948i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 - 2.14T + 7T^{2} \)
11 \( 1 + (0.139 - 0.139i)T - 11iT^{2} \)
13 \( 1 + (-1.17 + 4.38i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.66 - 4.61i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.10 + 1.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.437 - 1.63i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + 1.42T + 31T^{2} \)
37 \( 1 + (-1.34 + 1.34i)T - 37iT^{2} \)
41 \( 1 + (2.88 - 5.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.75 - 6.56i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.52 - 1.45i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.83 + 1.56i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.45 - 1.19i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (10.6 + 2.84i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (4.27 + 1.14i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-7.12 - 4.11i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.11 + 2.37i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.36 + 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.21 + 2.21i)T + 83iT^{2} \)
89 \( 1 + (7.09 + 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.61 + 3.24i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.02782997707228776310504541878, −10.75607086995488694221256787332, −8.420711311502241805767778112206, −8.120817601480094595643321653246, −7.44637920711351306162881301071, −6.20976678540904645155935502807, −4.56434979922149152422702720950, −3.09220554861628483206390934032, −1.63396843394458592502642693697, −0.05675442562266169729233765167, 3.93386997861237572327426771482, 4.48792631617153532681441203878, 5.20374176651445631499018458689, 6.67175272821944437629926070520, 7.964284845229216554583898352655, 8.839512429544948866733433630171, 9.366894872396532659736403681706, 10.74137655912043438659307588526, 11.26705390763965349791732479632, 12.00956171163579944943088711827

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