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Properties

Label 2-354-59.58-c2-0-0
Degree $2$
Conductor $354$
Sign $-0.803 + 0.595i$
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 + 0.383·5-s − 2.44i·6-s + 1.10·7-s − 2.82i·8-s + 2.99·9-s + 0.542i·10-s + 6.78i·11-s + 3.46·12-s + 11.1i·13-s + 1.56i·14-s − 0.664·15-s + 4.00·16-s − 23.0·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.500·4-s + 0.0766·5-s − 0.408i·6-s + 0.158·7-s − 0.353i·8-s + 0.333·9-s + 0.0542i·10-s + 0.617i·11-s + 0.288·12-s + 0.860i·13-s + 0.111i·14-s − 0.0442·15-s + 0.250·16-s − 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.803 + 0.595i$
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: \((2,\ 354,\ (\ :1),\ -0.803 + 0.595i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0699169 - 0.211905i\)
\(L(\frac12)\) \(\approx\) \(0.0699169 - 0.211905i\)
\(L(2)\) 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 \( 1 + (35.1 + 47.4i)T \)
good5 \( 1 - 0.383T + 25T^{2} \)
7 \( 1 - 1.10T + 49T^{2} \)
11 \( 1 - 6.78iT - 121T^{2} \)
13 \( 1 - 11.1iT - 169T^{2} \)
17 \( 1 + 23.0T + 289T^{2} \)
19 \( 1 + 25.1T + 361T^{2} \)
23 \( 1 + 33.4iT - 529T^{2} \)
29 \( 1 - 25.0T + 841T^{2} \)
31 \( 1 + 15.6iT - 961T^{2} \)
37 \( 1 + 15.7iT - 1.36e3T^{2} \)
41 \( 1 + 77.9T + 1.68e3T^{2} \)
43 \( 1 - 78.2iT - 1.84e3T^{2} \)
47 \( 1 + 22.0iT - 2.20e3T^{2} \)
53 \( 1 + 9.99T + 2.80e3T^{2} \)
61 \( 1 - 38.9iT - 3.72e3T^{2} \)
67 \( 1 - 0.743iT - 4.48e3T^{2} \)
71 \( 1 + 34.1T + 5.04e3T^{2} \)
73 \( 1 - 56.2iT - 5.32e3T^{2} \)
79 \( 1 - 112.T + 6.24e3T^{2} \)
83 \( 1 + 92.0iT - 6.88e3T^{2} \)
89 \( 1 - 133. iT - 7.92e3T^{2} \)
97 \( 1 - 120. iT - 9.40e3T^{2} \)
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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

−11.81139615570407528424659009528, −10.86517583552597685575596997809, −9.931559185444431410971866881414, −8.913160385792391972401964575493, −8.025902017304829712671797709068, −6.65563826345469919768941232464, −6.39509599956375508945711545486, −4.82646821799996210080702119803, −4.24070178719217227926576483564, −2.06574568624764923984462725239, 0.10474827922035000623532042437, 1.83623159043820537163270660804, 3.37646363759784979693281473431, 4.60916692918552120034719995954, 5.65612280273657489559666690939, 6.69597446269525771398644002582, 8.075402248979765772632605702015, 8.912806961265786992009253394716, 10.08809795088033886775748113195, 10.78058174350257935397572858999

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