Properties

Label 2-354-59.58-c2-0-9
Degree $2$
Conductor $354$
Sign $0.213 - 0.976i$
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 + 0.969·5-s + 2.44i·6-s + 3.05·7-s − 2.82i·8-s + 2.99·9-s + 1.37i·10-s + 3.80i·11-s − 3.46·12-s + 11.7i·13-s + 4.31i·14-s + 1.67·15-s + 4.00·16-s + 18.5·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577·3-s − 0.500·4-s + 0.193·5-s + 0.408i·6-s + 0.436·7-s − 0.353i·8-s + 0.333·9-s + 0.137i·10-s + 0.345i·11-s − 0.288·12-s + 0.901i·13-s + 0.308i·14-s + 0.111·15-s + 0.250·16-s + 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.213 - 0.976i$
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: \((2,\ 354,\ (\ :1),\ 0.213 - 0.976i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.65418 + 1.33127i\)
\(L(\frac12)\) \(\approx\) \(1.65418 + 1.33127i\)
\(L(2)\) 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 \( 1 + (57.6 + 12.6i)T \)
good5 \( 1 - 0.969T + 25T^{2} \)
7 \( 1 - 3.05T + 49T^{2} \)
11 \( 1 - 3.80iT - 121T^{2} \)
13 \( 1 - 11.7iT - 169T^{2} \)
17 \( 1 - 18.5T + 289T^{2} \)
19 \( 1 - 19.0T + 361T^{2} \)
23 \( 1 - 12.7iT - 529T^{2} \)
29 \( 1 - 33.5T + 841T^{2} \)
31 \( 1 - 34.8iT - 961T^{2} \)
37 \( 1 - 5.62iT - 1.36e3T^{2} \)
41 \( 1 - 2.32T + 1.68e3T^{2} \)
43 \( 1 + 3.84iT - 1.84e3T^{2} \)
47 \( 1 + 53.4iT - 2.20e3T^{2} \)
53 \( 1 - 9.70T + 2.80e3T^{2} \)
61 \( 1 - 6.26iT - 3.72e3T^{2} \)
67 \( 1 - 46.7iT - 4.48e3T^{2} \)
71 \( 1 - 58.6T + 5.04e3T^{2} \)
73 \( 1 - 3.41iT - 5.32e3T^{2} \)
79 \( 1 + 44.9T + 6.24e3T^{2} \)
83 \( 1 + 147. iT - 6.88e3T^{2} \)
89 \( 1 + 92.3iT - 7.92e3T^{2} \)
97 \( 1 - 123. iT - 9.40e3T^{2} \)
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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

−11.60665209474100665735993590698, −10.17100904034120931397209966920, −9.518349426473145342658556348354, −8.532190483667923646077237053050, −7.66801851845902377409157537269, −6.85239323760158370868387899086, −5.59382559497769487737365124719, −4.57301209923642831049590876443, −3.29820582863924633883862730548, −1.56275192314492399966972421570, 1.07320145533132924775610559121, 2.60438773195877667829220597791, 3.61334211424298934269157938512, 4.94108157685874974626329950250, 6.01517585767190539996784109758, 7.68712156058265996613508585026, 8.205426742679448145098039202356, 9.424949610445760697594984203571, 10.07273716331331368530101579953, 11.01930829248658206313444078066

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