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Properties

Label 2-304-16.13-c1-0-27
Degree $2$
Conductor $304$
Sign $-0.382 + 0.923i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (−0.292 − 0.292i)3-s − 2.00·4-s + (−2 + 2i)5-s + (0.414 − 0.414i)6-s − 1.58i·7-s − 2.82i·8-s − 2.82i·9-s + (−2.82 − 2.82i)10-s + (−4.41 + 4.41i)11-s + (0.585 + 0.585i)12-s + (−3.53 − 3.53i)13-s + 2.24·14-s + 1.17·15-s + 4.00·16-s + 1.82·17-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−0.169 − 0.169i)3-s − 1.00·4-s + (−0.894 + 0.894i)5-s + (0.169 − 0.169i)6-s − 0.599i·7-s − 1.00i·8-s − 0.942i·9-s + (−0.894 − 0.894i)10-s + (−1.33 + 1.33i)11-s + (0.169 + 0.169i)12-s + (−0.980 − 0.980i)13-s + 0.599·14-s + 0.302·15-s + 1.00·16-s + 0.443·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.382 + 0.923i$
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: \(1\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
19 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.292 + 0.292i)T + 3iT^{2} \)
5 \( 1 + (2 - 2i)T - 5iT^{2} \)
7 \( 1 + 1.58iT - 7T^{2} \)
11 \( 1 + (4.41 - 4.41i)T - 11iT^{2} \)
13 \( 1 + (3.53 + 3.53i)T + 13iT^{2} \)
17 \( 1 - 1.82T + 17T^{2} \)
23 \( 1 + 0.414iT - 23T^{2} \)
29 \( 1 + (3.29 + 3.29i)T + 29iT^{2} \)
31 \( 1 + 6.58T + 31T^{2} \)
37 \( 1 + (-2 + 2i)T - 37iT^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (4.82 - 4.82i)T - 43iT^{2} \)
47 \( 1 + 9.65T + 47T^{2} \)
53 \( 1 + (-0.949 + 0.949i)T - 53iT^{2} \)
59 \( 1 + (0.0502 - 0.0502i)T - 59iT^{2} \)
61 \( 1 + (-4.24 - 4.24i)T + 61iT^{2} \)
67 \( 1 + (-3.70 - 3.70i)T + 67iT^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + 1.82iT - 73T^{2} \)
79 \( 1 + 3.41T + 79T^{2} \)
83 \( 1 + (-1.41 - 1.41i)T + 83iT^{2} \)
89 \( 1 + 16.4iT - 89T^{2} \)
97 \( 1 + 3.07T + 97T^{2} \)
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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

−11.45346096447575224016362827540, −10.21991098572274765392977383822, −9.705449360749446103050059172783, −8.031565778882431370333602159822, −7.43715506722137758201067683914, −6.88519031130137201748750881782, −5.51125211073012194691356946745, −4.36393640425192026390337413247, −3.13379495781981102126815402513, 0, 2.20155972756555981869520871231, 3.63650810110558894768217655940, 4.95925535405190860598283164858, 5.42911134053335820080527204605, 7.66056209021462832752579927214, 8.390199994645727583654843129682, 9.180634006718198706876426048084, 10.39015553976220960341310635585, 11.17660415884747551468644147194

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