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Properties

Label 2-354-3.2-c2-0-23
Degree $2$
Conductor $354$
Sign $0.894 + 0.447i$
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 + (2.68 + 1.34i)3-s − 2.00·4-s − 2.14i·5-s + (1.89 − 3.79i)6-s + 6.97·7-s + 2.82i·8-s + (5.39 + 7.20i)9-s − 3.03·10-s + 7.24i·11-s + (−5.36 − 2.68i)12-s + 14.9·13-s − 9.86i·14-s + (2.88 − 5.76i)15-s + 4.00·16-s + 13.0i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.894 + 0.447i)3-s − 0.500·4-s − 0.429i·5-s + (0.316 − 0.632i)6-s + 0.996·7-s + 0.353i·8-s + (0.599 + 0.800i)9-s − 0.303·10-s + 0.658i·11-s + (−0.447 − 0.223i)12-s + 1.14·13-s − 0.704i·14-s + (0.192 − 0.384i)15-s + 0.250·16-s + 0.768i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(9.64580\)
Root analytic conductor: \(3.10576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1),\ 0.894 + 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.39268 - 0.564895i\)
\(L(\frac12)\) \(\approx\) \(2.39268 - 0.564895i\)
\(L(2)\) 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 \( 1 + (-2.68 - 1.34i)T \)
59 \( 1 + 7.68iT \)
good5 \( 1 + 2.14iT - 25T^{2} \)
7 \( 1 - 6.97T + 49T^{2} \)
11 \( 1 - 7.24iT - 121T^{2} \)
13 \( 1 - 14.9T + 169T^{2} \)
17 \( 1 - 13.0iT - 289T^{2} \)
19 \( 1 + 24.8T + 361T^{2} \)
23 \( 1 + 34.0iT - 529T^{2} \)
29 \( 1 + 44.6iT - 841T^{2} \)
31 \( 1 - 46.3T + 961T^{2} \)
37 \( 1 + 35.4T + 1.36e3T^{2} \)
41 \( 1 - 35.3iT - 1.68e3T^{2} \)
43 \( 1 - 81.6T + 1.84e3T^{2} \)
47 \( 1 - 70.1iT - 2.20e3T^{2} \)
53 \( 1 + 35.4iT - 2.80e3T^{2} \)
61 \( 1 + 42.7T + 3.72e3T^{2} \)
67 \( 1 + 131.T + 4.48e3T^{2} \)
71 \( 1 + 128. iT - 5.04e3T^{2} \)
73 \( 1 + 67.8T + 5.32e3T^{2} \)
79 \( 1 + 31.1T + 6.24e3T^{2} \)
83 \( 1 + 30.8iT - 6.88e3T^{2} \)
89 \( 1 - 148. iT - 7.92e3T^{2} \)
97 \( 1 + 118.T + 9.40e3T^{2} \)
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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

−10.84694281268263473814892313026, −10.49153768281946728801678453583, −9.266806351863993633146657753770, −8.431617230694487434239277629406, −8.003081595333329483402141730919, −6.26993003891555876811583912473, −4.56427196180151309830569721670, −4.27306036264440099963114743678, −2.62649094366274393088717522303, −1.47147971551945579200309692268, 1.35455482132306218757064455892, 3.03351470994314826693190550097, 4.22550281172651854348911663952, 5.62057676477829573581201525856, 6.75280036376346653487826907640, 7.53171691546722181606955955179, 8.607690666680084009238035935325, 8.869495536100473678841337598664, 10.38744020444500239214788635215, 11.25351631089315828669624622912

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