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Properties

Label 2-351-117.25-c1-0-0
Degree $2$
Conductor $351$
Sign $-0.423 - 0.906i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + (−0.929 − 0.536i)2-s + (−0.423 − 0.733i)4-s + (−1.10 + 0.638i)5-s + (0.890 + 0.514i)7-s + 3.05i·8-s + 1.37·10-s + (−4.03 − 2.33i)11-s + (−3.55 + 0.600i)13-s + (−0.552 − 0.956i)14-s + (0.794 − 1.37i)16-s + 0.476·17-s + 6.69i·19-s + (0.936 + 0.540i)20-s + (2.50 + 4.33i)22-s + (0.479 + 0.831i)23-s + ⋯
L(s)  = 1  + (−0.657 − 0.379i)2-s + (−0.211 − 0.366i)4-s + (−0.494 + 0.285i)5-s + (0.336 + 0.194i)7-s + 1.08i·8-s + 0.433·10-s + (−1.21 − 0.702i)11-s + (−0.986 + 0.166i)13-s + (−0.147 − 0.255i)14-s + (0.198 − 0.344i)16-s + 0.115·17-s + 1.53i·19-s + (0.209 + 0.120i)20-s + (0.533 + 0.924i)22-s + (0.100 + 0.173i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.423 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.423 - 0.906i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.423 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117472 + 0.184502i\)
\(L(\frac12)\) \(\approx\) \(0.117472 + 0.184502i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (3.55 - 0.600i)T \)
good2 \( 1 + (0.929 + 0.536i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.10 - 0.638i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.890 - 0.514i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.03 + 2.33i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.476T + 17T^{2} \)
19 \( 1 - 6.69iT - 19T^{2} \)
23 \( 1 + (-0.479 - 0.831i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.68 - 8.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.66 - 0.963i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.94iT - 37T^{2} \)
41 \( 1 + (-1.31 + 0.762i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.31 - 2.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.92 + 3.41i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.582T + 53T^{2} \)
59 \( 1 + (-3.64 + 2.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.71 - 8.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.01 - 1.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.35iT - 71T^{2} \)
73 \( 1 + 12.8iT - 73T^{2} \)
79 \( 1 + (-6.45 + 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.86 - 5.11i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.85iT - 89T^{2} \)
97 \( 1 + (14.9 + 8.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.49152397624065668645651628686, −10.75542026643514280065853469173, −10.06034660388519832367381316817, −9.063181052943925197099183076629, −8.096241378932426767703891476435, −7.41871841394300432636313539799, −5.73877545041933829067682532198, −5.01850965765976278519980519247, −3.36340126708138891413867748308, −1.88642272930566346513235241934, 0.18002228307969266956766681449, 2.62691993294349135361846525543, 4.25661766999220990102497261482, 5.06591097159171828514152184324, 6.76701545641570275816453066077, 7.74587869559065440233256490369, 8.055584350032611075305931818504, 9.338919298396855173348786233301, 9.986431198611064166635422483219, 11.13066272786578468652982557649

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