Properties

Label 2-304-304.213-c1-0-7
Degree $2$
Conductor $304$
Sign $0.218 + 0.975i$
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  + (−1.07 − 0.922i)2-s + (−2.52 − 1.76i)3-s + (0.297 + 1.97i)4-s + (0.172 + 0.0150i)5-s + (1.07 + 4.21i)6-s + (3.01 + 1.74i)7-s + (1.50 − 2.39i)8-s + (2.21 + 6.08i)9-s + (−0.170 − 0.174i)10-s + (0.624 − 2.33i)11-s + (2.74 − 5.51i)12-s + (1.34 + 1.92i)13-s + (−1.62 − 4.64i)14-s + (−0.407 − 0.341i)15-s + (−3.82 + 1.17i)16-s + (5.11 + 1.86i)17-s + ⋯
L(s)  = 1  + (−0.757 − 0.652i)2-s + (−1.45 − 1.01i)3-s + (0.148 + 0.988i)4-s + (0.0769 + 0.00673i)5-s + (0.438 + 1.72i)6-s + (1.13 + 0.657i)7-s + (0.532 − 0.846i)8-s + (0.738 + 2.02i)9-s + (−0.0539 − 0.0552i)10-s + (0.188 − 0.702i)11-s + (0.791 − 1.59i)12-s + (0.373 + 0.533i)13-s + (−0.434 − 1.24i)14-s + (−0.105 − 0.0882i)15-s + (−0.955 + 0.294i)16-s + (1.24 + 0.451i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.218 + 0.975i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.218 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.512027 - 0.410244i\)
\(L(\frac12)\) \(\approx\) \(0.512027 - 0.410244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 + 0.922i)T \)
19 \( 1 + (4.29 + 0.741i)T \)
good3 \( 1 + (2.52 + 1.76i)T + (1.02 + 2.81i)T^{2} \)
5 \( 1 + (-0.172 - 0.0150i)T + (4.92 + 0.868i)T^{2} \)
7 \( 1 + (-3.01 - 1.74i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.624 + 2.33i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.34 - 1.92i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-5.11 - 1.86i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-3.02 + 3.60i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-1.31 + 2.80i)T + (-18.6 - 22.2i)T^{2} \)
31 \( 1 + (-3.56 + 6.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.65 - 7.65i)T + 37iT^{2} \)
41 \( 1 + (-1.71 + 0.302i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-4.19 - 0.367i)T + (42.3 + 7.46i)T^{2} \)
47 \( 1 + (1.06 - 0.388i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (0.862 + 9.85i)T + (-52.1 + 9.20i)T^{2} \)
59 \( 1 + (-4.13 - 8.87i)T + (-37.9 + 45.1i)T^{2} \)
61 \( 1 + (-4.04 + 0.354i)T + (60.0 - 10.5i)T^{2} \)
67 \( 1 + (-1.57 + 3.38i)T + (-43.0 - 51.3i)T^{2} \)
71 \( 1 + (3.37 + 4.02i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + (-10.1 + 1.78i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.40 - 7.99i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.233 - 0.870i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (11.9 + 2.10i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (13.5 + 4.94i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.43351165934819285339285714796, −11.03445924941255871170717732674, −9.901069596364723858716418018640, −8.400393473478623156901195203825, −7.914744796629059996381731519229, −6.58739291668197253838742186672, −5.77805788952250548192744934535, −4.41411129633436809218668958748, −2.22279078806759732604496005992, −1.01609365943535487323314858281, 1.13058486895534614679783565429, 4.18987346701190520947818417604, 5.11987240353211369955246735891, 5.84871097323553393028556533505, 7.05523054607824213670564061174, 8.006304611022298324225588690978, 9.365834787031487378088562768385, 10.14415871353594252912290302554, 10.84553896440505164855650776569, 11.41332669886927713356329852569

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