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Properties

Label 2-304-16.13-c1-0-9
Degree $2$
Conductor $304$
Sign $0.824 + 0.566i$
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  + (−0.472 − 1.33i)2-s + (−0.194 − 0.194i)3-s + (−1.55 + 1.25i)4-s + (−1.01 + 1.01i)5-s + (−0.167 + 0.350i)6-s + 2.78i·7-s + (2.41 + 1.47i)8-s − 2.92i·9-s + (1.84 + 0.877i)10-s + (4.15 − 4.15i)11-s + (0.546 + 0.0574i)12-s + (3.85 + 3.85i)13-s + (3.71 − 1.31i)14-s + 0.396·15-s + (0.831 − 3.91i)16-s + 4.48·17-s + ⋯
L(s)  = 1  + (−0.333 − 0.942i)2-s + (−0.112 − 0.112i)3-s + (−0.777 + 0.629i)4-s + (−0.455 + 0.455i)5-s + (−0.0682 + 0.143i)6-s + 1.05i·7-s + (0.852 + 0.522i)8-s − 0.974i·9-s + (0.581 + 0.277i)10-s + (1.25 − 1.25i)11-s + (0.157 + 0.0165i)12-s + (1.07 + 1.07i)13-s + (0.994 − 0.352i)14-s + 0.102·15-s + (0.207 − 0.978i)16-s + 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.824 + 0.566i$
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: \((2,\ 304,\ (\ :1/2),\ 0.824 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966913 - 0.300172i\)
\(L(\frac12)\) \(\approx\) \(0.966913 - 0.300172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.472 + 1.33i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.194 + 0.194i)T + 3iT^{2} \)
5 \( 1 + (1.01 - 1.01i)T - 5iT^{2} \)
7 \( 1 - 2.78iT - 7T^{2} \)
11 \( 1 + (-4.15 + 4.15i)T - 11iT^{2} \)
13 \( 1 + (-3.85 - 3.85i)T + 13iT^{2} \)
17 \( 1 - 4.48T + 17T^{2} \)
23 \( 1 - 3.66iT - 23T^{2} \)
29 \( 1 + (-2.04 - 2.04i)T + 29iT^{2} \)
31 \( 1 + 2.97T + 31T^{2} \)
37 \( 1 + (-8.39 + 8.39i)T - 37iT^{2} \)
41 \( 1 - 5.02iT - 41T^{2} \)
43 \( 1 + (5.43 - 5.43i)T - 43iT^{2} \)
47 \( 1 + 6.73T + 47T^{2} \)
53 \( 1 + (-6.12 + 6.12i)T - 53iT^{2} \)
59 \( 1 + (-6.65 + 6.65i)T - 59iT^{2} \)
61 \( 1 + (2.39 + 2.39i)T + 61iT^{2} \)
67 \( 1 + (8.65 + 8.65i)T + 67iT^{2} \)
71 \( 1 + 3.99iT - 71T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 + 8.69T + 79T^{2} \)
83 \( 1 + (-8.43 - 8.43i)T + 83iT^{2} \)
89 \( 1 - 4.30iT - 89T^{2} \)
97 \( 1 + 4.86T + 97T^{2} \)
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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

−11.42842646449641574049604055647, −11.23279365991189930998782733867, −9.524165196589490550080226612027, −9.058533571882442766436927907330, −8.172373007806974094420575045097, −6.70209286714158382125661837479, −5.71478277915147418065864104482, −3.85776035713694450506076040507, −3.24519044863444308502800382535, −1.33829630814000039144338039734, 1.13811358221418576889732980753, 3.97090320148442287492115931556, 4.70182549591259437087715479383, 5.98815701717986953739887475289, 7.13983227464333916046663835523, 7.908435084053078776875596977537, 8.690867118328836339809637989700, 10.09489398797109262478288979537, 10.41372050953029326268984973514, 11.81504235620689180015492806759

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