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Properties

Label 2-351-39.11-c1-0-16
Degree $2$
Conductor $351$
Sign $0.967 - 0.254i$
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  + (2.17 + 0.583i)2-s + (2.66 + 1.54i)4-s + (1.66 − 1.66i)5-s + (−0.0615 − 0.229i)7-s + (1.72 + 1.72i)8-s + (4.60 − 2.65i)10-s + (0.174 − 0.650i)11-s + (−3.51 + 0.782i)13-s − 0.536i·14-s + (−0.335 − 0.581i)16-s + (−3.30 + 5.72i)17-s + (−0.752 + 0.201i)19-s + (7.01 − 1.88i)20-s + (0.759 − 1.31i)22-s + (3.22 + 5.57i)23-s + ⋯
L(s)  = 1  + (1.53 + 0.412i)2-s + (1.33 + 0.770i)4-s + (0.745 − 0.745i)5-s + (−0.0232 − 0.0868i)7-s + (0.608 + 0.608i)8-s + (1.45 − 0.840i)10-s + (0.0525 − 0.196i)11-s + (−0.976 + 0.216i)13-s − 0.143i·14-s + (−0.0838 − 0.145i)16-s + (−0.801 + 1.38i)17-s + (−0.172 + 0.0462i)19-s + (1.56 − 0.420i)20-s + (0.161 − 0.280i)22-s + (0.671 + 1.16i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.967 - 0.254i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.967 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.06772 + 0.396936i\)
\(L(\frac12)\) \(\approx\) \(3.06772 + 0.396936i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.51 - 0.782i)T \)
good2 \( 1 + (-2.17 - 0.583i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.66 + 1.66i)T - 5iT^{2} \)
7 \( 1 + (0.0615 + 0.229i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.174 + 0.650i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (3.30 - 5.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.752 - 0.201i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.22 - 5.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.06 + 2.34i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.22 + 1.22i)T + 31iT^{2} \)
37 \( 1 + (9.00 + 2.41i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-8.51 - 2.28i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (8.68 + 5.01i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.19 + 2.19i)T + 47iT^{2} \)
53 \( 1 - 3.84iT - 53T^{2} \)
59 \( 1 + (-5.65 + 1.51i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.48 - 7.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.26 + 12.1i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.63 + 13.5i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.87 - 1.87i)T - 73iT^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 + (4.09 - 4.09i)T - 83iT^{2} \)
89 \( 1 + (2.19 - 8.19i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-5.37 + 1.43i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.98671361496718990874979868173, −10.82845479466966602573844957123, −9.642366425517030662366435942313, −8.706986471267599464788295188947, −7.35745442255118318140426758803, −6.36834436554061678361448628108, −5.48160634728342683744836678777, −4.71232100927107788238175172628, −3.60839237374002707001228278334, −2.05128603642592361665463762619, 2.32873899136681038935782825066, 2.97840849832895732148558187565, 4.54974277243598217711067786311, 5.26238625294469640649478924314, 6.49249931435517993749774244215, 7.05984924723217794356732037394, 8.791876966521209652562249196046, 9.976078769067693935558998399484, 10.74944928569925983492844253388, 11.64405926440242116558544478144

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