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Properties

Label 2-354-59.58-c2-0-6
Degree $2$
Conductor $354$
Sign $-0.324 - 0.945i$
Analytic cond. $9.64580$
Root an. cond. $3.10576$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 1.73·3-s − 2.00·4-s + 7.71·5-s − 2.44i·6-s − 7.56·7-s − 2.82i·8-s + 2.99·9-s + 10.9i·10-s − 2.42i·11-s + 3.46·12-s + 20.7i·13-s − 10.6i·14-s − 13.3·15-s + 4.00·16-s + 31.7·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.500·4-s + 1.54·5-s − 0.408i·6-s − 1.08·7-s − 0.353i·8-s + 0.333·9-s + 1.09i·10-s − 0.220i·11-s + 0.288·12-s + 1.59i·13-s − 0.764i·14-s − 0.890·15-s + 0.250·16-s + 1.86·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.324 - 0.945i$
Analytic conductor: \(9.64580\)
Root analytic conductor: \(3.10576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1),\ -0.324 - 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.822802 + 1.15256i\)
\(L(\frac12)\) \(\approx\) \(0.822802 + 1.15256i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + 1.73T \)
59 \( 1 + (-55.8 + 19.1i)T \)
good5 \( 1 - 7.71T + 25T^{2} \)
7 \( 1 + 7.56T + 49T^{2} \)
11 \( 1 + 2.42iT - 121T^{2} \)
13 \( 1 - 20.7iT - 169T^{2} \)
17 \( 1 - 31.7T + 289T^{2} \)
19 \( 1 + 4.95T + 361T^{2} \)
23 \( 1 - 31.3iT - 529T^{2} \)
29 \( 1 + 11.8T + 841T^{2} \)
31 \( 1 - 8.07iT - 961T^{2} \)
37 \( 1 - 41.8iT - 1.36e3T^{2} \)
41 \( 1 + 44.4T + 1.68e3T^{2} \)
43 \( 1 - 6.57iT - 1.84e3T^{2} \)
47 \( 1 - 92.7iT - 2.20e3T^{2} \)
53 \( 1 - 45.1T + 2.80e3T^{2} \)
61 \( 1 - 42.6iT - 3.72e3T^{2} \)
67 \( 1 + 76.9iT - 4.48e3T^{2} \)
71 \( 1 + 32.2T + 5.04e3T^{2} \)
73 \( 1 + 119. iT - 5.32e3T^{2} \)
79 \( 1 - 94.3T + 6.24e3T^{2} \)
83 \( 1 + 108. iT - 6.88e3T^{2} \)
89 \( 1 + 58.7iT - 7.92e3T^{2} \)
97 \( 1 + 21.2iT - 9.40e3T^{2} \)
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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

−11.61802197677184163210580866575, −10.19288319354821724769208247711, −9.693726882535690019282640836112, −9.025290997034723798581559234385, −7.45743954735095953507497294953, −6.41356286108758279828149059040, −5.96701287564747890744956790001, −4.98949186703300296446774000442, −3.38973363788825400469421002707, −1.52311747821546402194618183535, 0.74169504242558658180679231161, 2.36644816698630669289422534825, 3.53018112117557756694563391068, 5.35437446676898288161736368333, 5.76421303195024657153414538026, 6.92039976446858559549861743281, 8.392591252399674648318765342386, 9.665825038814737967492881728860, 10.11959066125830107481946614147, 10.59371063677528071695747491796

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