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Properties

Label 2-354-177.176-c3-0-47
Degree $2$
Conductor $354$
Sign $-0.0408 + 0.999i$
Analytic cond. $20.8866$
Root an. cond. $4.57019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + (2.99 + 4.24i)3-s + 4·4-s − 19.3i·5-s + (−5.98 − 8.49i)6-s + 6.85·7-s − 8·8-s + (−9.10 + 25.4i)9-s + 38.6i·10-s − 4.18·11-s + (11.9 + 16.9i)12-s − 24.0i·13-s − 13.7·14-s + (82.0 − 57.7i)15-s + 16·16-s − 81.9i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.575 + 0.817i)3-s + 0.5·4-s − 1.72i·5-s + (−0.407 − 0.578i)6-s + 0.370·7-s − 0.353·8-s + (−0.337 + 0.941i)9-s + 1.22i·10-s − 0.114·11-s + (0.287 + 0.408i)12-s − 0.513i·13-s − 0.261·14-s + (1.41 − 0.994i)15-s + 0.250·16-s − 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0408 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0408 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.0408 + 0.999i$
Analytic conductor: \(20.8866\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :3/2),\ -0.0408 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.253697711\)
\(L(\frac12)\) \(\approx\) \(1.253697711\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 + (-2.99 - 4.24i)T \)
59 \( 1 + (-359. - 275. i)T \)
good5 \( 1 + 19.3iT - 125T^{2} \)
7 \( 1 - 6.85T + 343T^{2} \)
11 \( 1 + 4.18T + 1.33e3T^{2} \)
13 \( 1 + 24.0iT - 2.19e3T^{2} \)
17 \( 1 + 81.9iT - 4.91e3T^{2} \)
19 \( 1 - 122.T + 6.85e3T^{2} \)
23 \( 1 + 216.T + 1.21e4T^{2} \)
29 \( 1 + 136. iT - 2.43e4T^{2} \)
31 \( 1 - 342. iT - 2.97e4T^{2} \)
37 \( 1 + 346. iT - 5.06e4T^{2} \)
41 \( 1 + 280. iT - 6.89e4T^{2} \)
43 \( 1 + 353. iT - 7.95e4T^{2} \)
47 \( 1 + 88.0T + 1.03e5T^{2} \)
53 \( 1 + 89.3iT - 1.48e5T^{2} \)
61 \( 1 + 198. iT - 2.26e5T^{2} \)
67 \( 1 + 856. iT - 3.00e5T^{2} \)
71 \( 1 - 384. iT - 3.57e5T^{2} \)
73 \( 1 + 586. iT - 3.89e5T^{2} \)
79 \( 1 - 417.T + 4.93e5T^{2} \)
83 \( 1 - 1.26e3T + 5.71e5T^{2} \)
89 \( 1 - 329.T + 7.04e5T^{2} \)
97 \( 1 - 49.9iT - 9.12e5T^{2} \)
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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

−10.49100261070667795217966052299, −9.615370631462079842132867037985, −9.042788727043410355526107717740, −8.198769151854884273440840207941, −7.55914598588295856645773643319, −5.53713884312923654831402783583, −4.92961745004767585417916959322, −3.63588987143886407326887598420, −2.00379208946859965956701723867, −0.50402295662734037370960365797, 1.61179767229924186699662177754, 2.66816855649253083141515286276, 3.72018079644094806969368308834, 6.04873308316580048319878221764, 6.64308572488989570352545491853, 7.73625117171162942633220560590, 8.075079299386834591711882733217, 9.572805374141250426308448491670, 10.18484809643485780612469267788, 11.42352101401501368453978493581

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