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Properties

Label 2-304-19.12-c4-0-26
Degree $2$
Conductor $304$
Sign $-0.156 + 0.987i$
Analytic cond. $31.4244$
Root an. cond. $5.60575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 + 1.11i)3-s + (−15.6 − 27.0i)5-s + 6.74·7-s + (−37.9 + 65.8i)9-s + 124.·11-s + (134. + 77.7i)13-s + (60.5 + 34.9i)15-s + (127. + 220. i)17-s + (−161. − 322. i)19-s + (−13.0 + 7.55i)21-s + (193. − 334. i)23-s + (−175. + 303. i)25-s − 351. i·27-s + (−985. − 569. i)29-s − 593. i·31-s + ⋯
L(s)  = 1  + (−0.215 + 0.124i)3-s + (−0.624 − 1.08i)5-s + 0.137·7-s + (−0.469 + 0.812i)9-s + 1.03·11-s + (0.796 + 0.460i)13-s + (0.269 + 0.155i)15-s + (0.440 + 0.763i)17-s + (−0.447 − 0.894i)19-s + (−0.0296 + 0.0171i)21-s + (0.365 − 0.632i)23-s + (−0.280 + 0.486i)25-s − 0.482i·27-s + (−1.17 − 0.676i)29-s − 0.617i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ -0.156 + 0.987i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.219971138\)
\(L(\frac12)\) \(\approx\) \(1.219971138\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (161. + 322. i)T \)
good3 \( 1 + (1.93 - 1.11i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (15.6 + 27.0i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 6.74T + 2.40e3T^{2} \)
11 \( 1 - 124.T + 1.46e4T^{2} \)
13 \( 1 + (-134. - 77.7i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-127. - 220. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-193. + 334. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (985. + 569. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 593. iT - 9.23e5T^{2} \)
37 \( 1 + 360. iT - 1.87e6T^{2} \)
41 \( 1 + (1.64e3 - 951. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (794. + 1.37e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.34e3 + 2.33e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (1.97 + 1.13i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-3.49e3 + 2.01e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-3.48e3 + 6.03e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-4.54e3 - 2.62e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (2.21e3 - 1.28e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-110. - 191. i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (2.24e3 - 1.29e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 7.43e3T + 4.74e7T^{2} \)
89 \( 1 + (8.82e3 + 5.09e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-8.25e3 + 4.76e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.17457436266190096634424459003, −9.820194092376553668266274257370, −8.617195118114107318604617965552, −8.333825608057695880289004600648, −6.89846635897693536232974287140, −5.70780091489298122994370960835, −4.63033022618402598617560735646, −3.78076853733077833771628538775, −1.84673041874803029862127338184, −0.43380756769927619030010580639, 1.22881108865937446307565820701, 3.17186637254465204325912888313, 3.80093245484730570522821483506, 5.55321296788177409280967863035, 6.54917750028790509012137300456, 7.30351233393540690395837750378, 8.465080705014251497657521570259, 9.455883696006971155180678382004, 10.61298250001804140283200536317, 11.44119228005476859324792842634

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