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Properties

Label 2-354-59.7-c1-0-1
Degree $2$
Conductor $354$
Sign $-0.121 - 0.992i$
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  + (−0.561 − 0.827i)2-s + (−0.0541 + 0.998i)3-s + (−0.370 + 0.928i)4-s + (−0.0493 + 0.0228i)5-s + (0.856 − 0.515i)6-s + (−0.676 + 2.43i)7-s + (0.976 − 0.214i)8-s + (−0.994 − 0.108i)9-s + (0.0466 + 0.0280i)10-s + (0.0375 − 0.0355i)11-s + (−0.907 − 0.419i)12-s + (−4.97 + 0.541i)13-s + (2.39 − 0.807i)14-s + (−0.0201 − 0.0505i)15-s + (−0.725 − 0.687i)16-s + (1.56 + 5.64i)17-s + ⋯
L(s)  = 1  + (−0.396 − 0.585i)2-s + (−0.0312 + 0.576i)3-s + (−0.185 + 0.464i)4-s + (−0.0220 + 0.0102i)5-s + (0.349 − 0.210i)6-s + (−0.255 + 0.921i)7-s + (0.345 − 0.0760i)8-s + (−0.331 − 0.0360i)9-s + (0.0147 + 0.00886i)10-s + (0.0113 − 0.0107i)11-s + (−0.261 − 0.121i)12-s + (−1.38 + 0.150i)13-s + (0.640 − 0.215i)14-s + (−0.00519 − 0.0130i)15-s + (−0.181 − 0.171i)16-s + (0.380 + 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.121 - 0.992i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ -0.121 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.473248 + 0.534638i\)
\(L(\frac12)\) \(\approx\) \(0.473248 + 0.534638i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.561 + 0.827i)T \)
3 \( 1 + (0.0541 - 0.998i)T \)
59 \( 1 + (-7.00 - 3.14i)T \)
good5 \( 1 + (0.0493 - 0.0228i)T + (3.23 - 3.81i)T^{2} \)
7 \( 1 + (0.676 - 2.43i)T + (-5.99 - 3.60i)T^{2} \)
11 \( 1 + (-0.0375 + 0.0355i)T + (0.595 - 10.9i)T^{2} \)
13 \( 1 + (4.97 - 0.541i)T + (12.6 - 2.79i)T^{2} \)
17 \( 1 + (-1.56 - 5.64i)T + (-14.5 + 8.76i)T^{2} \)
19 \( 1 + (3.72 - 2.83i)T + (5.08 - 18.3i)T^{2} \)
23 \( 1 + (-3.48 + 6.56i)T + (-12.9 - 19.0i)T^{2} \)
29 \( 1 + (4.33 - 6.39i)T + (-10.7 - 26.9i)T^{2} \)
31 \( 1 + (-8.10 - 6.16i)T + (8.29 + 29.8i)T^{2} \)
37 \( 1 + (-0.292 - 0.0644i)T + (33.5 + 15.5i)T^{2} \)
41 \( 1 + (-0.0856 - 0.161i)T + (-23.0 + 33.9i)T^{2} \)
43 \( 1 + (7.33 + 6.95i)T + (2.32 + 42.9i)T^{2} \)
47 \( 1 + (-1.94 - 0.899i)T + (30.4 + 35.8i)T^{2} \)
53 \( 1 + (-9.27 + 5.57i)T + (24.8 - 46.8i)T^{2} \)
61 \( 1 + (-1.89 - 2.79i)T + (-22.5 + 56.6i)T^{2} \)
67 \( 1 + (-2.89 + 0.636i)T + (60.8 - 28.1i)T^{2} \)
71 \( 1 + (3.69 + 1.70i)T + (45.9 + 54.1i)T^{2} \)
73 \( 1 + (9.37 - 3.15i)T + (58.1 - 44.1i)T^{2} \)
79 \( 1 + (0.382 + 7.06i)T + (-78.5 + 8.54i)T^{2} \)
83 \( 1 + (2.48 + 15.1i)T + (-78.6 + 26.5i)T^{2} \)
89 \( 1 + (-8.24 + 12.1i)T + (-32.9 - 82.6i)T^{2} \)
97 \( 1 + (1.43 + 0.483i)T + (77.2 + 58.7i)T^{2} \)
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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

−11.78934086408428135729377093226, −10.47827759499689321927845237951, −10.12761753969987514096312359220, −8.927796863271866212689617552531, −8.431995186906624080244789054526, −7.03664643913888331980210863881, −5.75322675796789292844909368751, −4.63531460200615266220049128428, −3.35570776906492384492051583144, −2.12011437953660174369738920164, 0.54356796652000740372815924528, 2.54611189256608940704957672698, 4.33065918330304557396445098704, 5.48296648693065072931366923098, 6.74035708978742273346557331168, 7.37875398923043495535754333392, 8.100299654381890900780544365820, 9.529806986118353857999594895888, 9.959479212489501896628363488139, 11.27816157791538490583555036251

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