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Properties

Label 2-351-117.25-c1-0-10
Degree $2$
Conductor $351$
Sign $-0.515 + 0.856i$
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  + (0.784 + 0.452i)2-s + (−0.589 − 1.02i)4-s + (−1.94 + 1.12i)5-s + (−2.97 − 1.71i)7-s − 2.87i·8-s − 2.03·10-s + (−3.20 − 1.84i)11-s + (0.351 − 3.58i)13-s + (−1.55 − 2.69i)14-s + (0.124 − 0.214i)16-s + 4.21·17-s + 4.25i·19-s + (2.29 + 1.32i)20-s + (−1.67 − 2.89i)22-s + (−1.89 − 3.27i)23-s + ⋯
L(s)  = 1  + (0.554 + 0.320i)2-s + (−0.294 − 0.510i)4-s + (−0.868 + 0.501i)5-s + (−1.12 − 0.649i)7-s − 1.01i·8-s − 0.642·10-s + (−0.965 − 0.557i)11-s + (0.0975 − 0.995i)13-s + (−0.415 − 0.720i)14-s + (0.0310 − 0.0537i)16-s + 1.02·17-s + 0.975i·19-s + (0.512 + 0.295i)20-s + (−0.356 − 0.618i)22-s + (−0.394 − 0.683i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.515 + 0.856i$
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: \((2,\ 351,\ (\ :1/2),\ -0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.334532 - 0.591509i\)
\(L(\frac12)\) \(\approx\) \(0.334532 - 0.591509i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.351 + 3.58i)T \)
good2 \( 1 + (-0.784 - 0.452i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.94 - 1.12i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.97 + 1.71i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.20 + 1.84i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.21T + 17T^{2} \)
19 \( 1 - 4.25iT - 19T^{2} \)
23 \( 1 + (1.89 + 3.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.18 + 2.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.37 - 3.67i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.49iT - 37T^{2} \)
41 \( 1 + (-6.86 + 3.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.450 - 0.779i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.80 - 2.77i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.59T + 53T^{2} \)
59 \( 1 + (4.44 - 2.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.50 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.7 + 6.75i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.65iT - 71T^{2} \)
73 \( 1 + 5.45iT - 73T^{2} \)
79 \( 1 + (5.46 - 9.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.465 + 0.268i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.75iT - 89T^{2} \)
97 \( 1 + (5.87 + 3.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.89052806767761926977660973083, −10.39067297499753277343817398400, −9.553298365248154027062140123700, −8.057041525908352322487220849222, −7.32221414283122967898612953956, −6.16587747929141334879441684422, −5.39563069847212672213784946644, −3.87955982687315163177543373015, −3.23994196067971516154054373590, −0.38327320351699233072575498916, 2.56748815259579748487335092194, 3.65787466478597023027815731487, 4.64944199627838093607359550861, 5.69369707281869709358772865717, 7.17267461699176377971963431559, 8.058445776308486321070974994990, 9.023689137212457321144248262711, 9.823655409984945848864015438241, 11.28276206861277301431372931371, 11.97658906039578654584241481263

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