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Properties

Label 2-352-11.9-c1-0-8
Degree $2$
Conductor $352$
Sign $0.734 + 0.678i$
Analytic cond. $2.81073$
Root an. cond. $1.67652$
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  + (2.57 − 1.87i)3-s + (−0.0283 − 0.0874i)5-s + (3.14 + 2.28i)7-s + (2.20 − 6.78i)9-s + (−3.27 + 0.500i)11-s + (−0.805 + 2.47i)13-s + (−0.236 − 0.171i)15-s + (−0.163 − 0.504i)17-s + (−4.06 + 2.95i)19-s + 12.3·21-s − 3.05·23-s + (4.03 − 2.93i)25-s + (−4.06 − 12.5i)27-s + (−5.34 − 3.88i)29-s + (−1.47 + 4.53i)31-s + ⋯
L(s)  = 1  + (1.48 − 1.08i)3-s + (−0.0127 − 0.0390i)5-s + (1.19 + 0.864i)7-s + (0.734 − 2.26i)9-s + (−0.988 + 0.150i)11-s + (−0.223 + 0.687i)13-s + (−0.0611 − 0.0443i)15-s + (−0.0397 − 0.122i)17-s + (−0.933 + 0.678i)19-s + 2.70·21-s − 0.636·23-s + (0.807 − 0.586i)25-s + (−0.782 − 2.40i)27-s + (−0.992 − 0.721i)29-s + (−0.264 + 0.814i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.734 + 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02442 - 0.791342i\)
\(L(\frac12)\) \(\approx\) \(2.02442 - 0.791342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (3.27 - 0.500i)T \)
good3 \( 1 + (-2.57 + 1.87i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.0283 + 0.0874i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-3.14 - 2.28i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.805 - 2.47i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.163 + 0.504i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.06 - 2.95i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.05T + 23T^{2} \)
29 \( 1 + (5.34 + 3.88i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.47 - 4.53i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.66 + 4.84i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.34 + 1.70i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 6.01T + 43T^{2} \)
47 \( 1 + (3.53 - 2.56i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.35 + 10.3i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.06 - 6.58i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.53 - 10.8i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 7.33T + 67T^{2} \)
71 \( 1 + (-3.46 - 10.6i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.04 - 1.48i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.66 + 8.20i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.473 - 1.45i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 3.93T + 89T^{2} \)
97 \( 1 + (1.48 - 4.56i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.66221442017156460609898284918, −10.35141445971035957159246265372, −9.069724362675089792483936191157, −8.463576341682315041494971877570, −7.81504626079003151407724987156, −6.91426001311571371182603606812, −5.55038932638310699944191144625, −4.09668470887958067930493218538, −2.49665290298949820042421704830, −1.86633797265422340031111796068, 2.15044964822075785785217920075, 3.38824234765292537873767353905, 4.45611834827589140717320956833, 5.23155032490116439013613324790, 7.33396413425900803322261747727, 8.021031122497946601016432343336, 8.689913094472272251557813794202, 9.755874116211685859133194030918, 10.71475063308009457638477482880, 10.96465492145610548877283126780

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