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Properties

Label 2-304-16.13-c1-0-0
Degree $2$
Conductor $304$
Sign $-0.0607 - 0.998i$
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  + (0.880 − 1.10i)2-s + (−1.52 − 1.52i)3-s + (−0.449 − 1.94i)4-s + (−2.81 + 2.81i)5-s + (−3.02 + 0.344i)6-s + 4.66i·7-s + (−2.55 − 1.21i)8-s + 1.64i·9-s + (0.637 + 5.59i)10-s + (0.655 − 0.655i)11-s + (−2.28 + 3.65i)12-s + (−3.00 − 3.00i)13-s + (5.15 + 4.10i)14-s + 8.58·15-s + (−3.59 + 1.75i)16-s − 4.83·17-s + ⋯
L(s)  = 1  + (0.622 − 0.782i)2-s + (−0.880 − 0.880i)3-s + (−0.224 − 0.974i)4-s + (−1.25 + 1.25i)5-s + (−1.23 + 0.140i)6-s + 1.76i·7-s + (−0.902 − 0.430i)8-s + 0.549i·9-s + (0.201 + 1.76i)10-s + (0.197 − 0.197i)11-s + (−0.659 + 1.05i)12-s + (−0.834 − 0.834i)13-s + (1.37 + 1.09i)14-s + 2.21·15-s + (−0.898 + 0.438i)16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0607 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0607 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.0607 - 0.998i$
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: \((2,\ 304,\ (\ :1/2),\ -0.0607 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.126748 + 0.134700i\)
\(L(\frac12)\) \(\approx\) \(0.126748 + 0.134700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.880 + 1.10i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.52 + 1.52i)T + 3iT^{2} \)
5 \( 1 + (2.81 - 2.81i)T - 5iT^{2} \)
7 \( 1 - 4.66iT - 7T^{2} \)
11 \( 1 + (-0.655 + 0.655i)T - 11iT^{2} \)
13 \( 1 + (3.00 + 3.00i)T + 13iT^{2} \)
17 \( 1 + 4.83T + 17T^{2} \)
23 \( 1 - 5.34iT - 23T^{2} \)
29 \( 1 + (1.34 + 1.34i)T + 29iT^{2} \)
31 \( 1 - 5.09T + 31T^{2} \)
37 \( 1 + (2.94 - 2.94i)T - 37iT^{2} \)
41 \( 1 - 2.93iT - 41T^{2} \)
43 \( 1 + (-4.63 + 4.63i)T - 43iT^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + (-1.10 + 1.10i)T - 53iT^{2} \)
59 \( 1 + (2.05 - 2.05i)T - 59iT^{2} \)
61 \( 1 + (3.46 + 3.46i)T + 61iT^{2} \)
67 \( 1 + (-1.54 - 1.54i)T + 67iT^{2} \)
71 \( 1 + 9.32iT - 71T^{2} \)
73 \( 1 + 4.60iT - 73T^{2} \)
79 \( 1 - 5.19T + 79T^{2} \)
83 \( 1 + (-5.51 - 5.51i)T + 83iT^{2} \)
89 \( 1 + 4.39iT - 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−11.83988757597958463580654402830, −11.48891186648873567459012309736, −10.64361991643508053119935076331, −9.309192740819643334059745738437, −7.986623602990810159920619888693, −6.76358614407718450465412110694, −6.07183071901464541310527047871, −4.97794188414003389270746592167, −3.33631883628640655392315993709, −2.32195189117443200389621386052, 0.11774116010972548062160301221, 3.99000385656899949313072427882, 4.42107962489922127962169633739, 4.92788157480173482602545065633, 6.60335930486486620953268880646, 7.45402219390259207119789531596, 8.422749248466857669644889543866, 9.536804133069614956878250004268, 10.80422965772329132748755727013, 11.56460853240754437384309674613

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