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Properties

Label 2-354-59.49-c1-0-3
Degree $2$
Conductor $354$
Sign $0.994 - 0.100i$
Analytic cond. $2.82670$
Root an. cond. $1.68128$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.370 − 0.928i)2-s + (0.994 + 0.108i)3-s + (−0.725 − 0.687i)4-s + (−2.09 + 2.46i)5-s + (0.468 − 0.883i)6-s + (3.22 + 1.93i)7-s + (−0.907 + 0.419i)8-s + (0.976 + 0.214i)9-s + (1.51 + 2.86i)10-s + (−0.0664 + 1.22i)11-s + (−0.647 − 0.762i)12-s + (3.78 − 0.833i)13-s + (2.99 − 2.27i)14-s + (−2.35 + 2.22i)15-s + (0.0541 + 0.998i)16-s + (1.60 − 0.965i)17-s + ⋯
L(s)  = 1  + (0.261 − 0.656i)2-s + (0.573 + 0.0624i)3-s + (−0.362 − 0.343i)4-s + (−0.938 + 1.10i)5-s + (0.191 − 0.360i)6-s + (1.21 + 0.732i)7-s + (−0.320 + 0.148i)8-s + (0.325 + 0.0716i)9-s + (0.480 + 0.905i)10-s + (−0.0200 + 0.369i)11-s + (−0.186 − 0.220i)12-s + (1.05 − 0.231i)13-s + (0.799 − 0.607i)14-s + (−0.607 + 0.575i)15-s + (0.0135 + 0.249i)16-s + (0.389 − 0.234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ 0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71612 + 0.0867481i\)
\(L(\frac12)\) \(\approx\) \(1.71612 + 0.0867481i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.370 + 0.928i)T \)
3 \( 1 + (-0.994 - 0.108i)T \)
59 \( 1 + (-2.23 - 7.34i)T \)
good5 \( 1 + (2.09 - 2.46i)T + (-0.808 - 4.93i)T^{2} \)
7 \( 1 + (-3.22 - 1.93i)T + (3.27 + 6.18i)T^{2} \)
11 \( 1 + (0.0664 - 1.22i)T + (-10.9 - 1.18i)T^{2} \)
13 \( 1 + (-3.78 + 0.833i)T + (11.7 - 5.45i)T^{2} \)
17 \( 1 + (-1.60 + 0.965i)T + (7.96 - 15.0i)T^{2} \)
19 \( 1 + (0.558 - 2.01i)T + (-16.2 - 9.79i)T^{2} \)
23 \( 1 + (1.75 + 2.58i)T + (-8.51 + 21.3i)T^{2} \)
29 \( 1 + (-1.74 - 4.37i)T + (-21.0 + 19.9i)T^{2} \)
31 \( 1 + (1.92 + 6.93i)T + (-26.5 + 15.9i)T^{2} \)
37 \( 1 + (4.48 + 2.07i)T + (23.9 + 28.1i)T^{2} \)
41 \( 1 + (2.48 - 3.66i)T + (-15.1 - 38.0i)T^{2} \)
43 \( 1 + (-0.269 - 4.97i)T + (-42.7 + 4.64i)T^{2} \)
47 \( 1 + (8.53 + 10.0i)T + (-7.60 + 46.3i)T^{2} \)
53 \( 1 + (-3.77 + 7.11i)T + (-29.7 - 43.8i)T^{2} \)
61 \( 1 + (-3.52 + 8.85i)T + (-44.2 - 41.9i)T^{2} \)
67 \( 1 + (-10.9 + 5.05i)T + (43.3 - 51.0i)T^{2} \)
71 \( 1 + (-2.28 - 2.69i)T + (-11.4 + 70.0i)T^{2} \)
73 \( 1 + (9.95 - 7.56i)T + (19.5 - 70.3i)T^{2} \)
79 \( 1 + (-8.31 + 0.904i)T + (77.1 - 16.9i)T^{2} \)
83 \( 1 + (-2.60 + 0.876i)T + (66.0 - 50.2i)T^{2} \)
89 \( 1 + (6.43 + 16.1i)T + (-64.6 + 61.2i)T^{2} \)
97 \( 1 + (3.99 + 3.04i)T + (25.9 + 93.4i)T^{2} \)
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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

−11.42813375388263328753059499367, −10.81771925519431097567845228064, −9.845406046937949018980559978511, −8.481689036584407711800226528687, −8.027406166638437046763162802487, −6.79513857732361074971896076939, −5.38071447182825251288726305828, −4.11151049940934872332068222887, −3.18819595990819502273939057865, −1.93292138274314949568266966718, 1.25173750431495913294000982523, 3.65403500585281093223476233944, 4.39830675867719844833048926780, 5.36332882713030888412109113760, 6.88896440361834585280634262911, 8.024564914138930679972044426896, 8.263149153105656426144387203505, 9.150059701939907126918529698838, 10.65495646453641447355207706926, 11.60014669259707496387019033031

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