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Properties

Label 2-349-349.348-c1-0-25
Degree $2$
Conductor $349$
Sign $-0.995 - 0.0957i$
Analytic cond. $2.78677$
Root an. cond. $1.66936$
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  − 2.26i·2-s + 0.674·3-s − 3.11·4-s − 0.0827·5-s − 1.52i·6-s − 1.39i·7-s + 2.51i·8-s − 2.54·9-s + 0.186i·10-s − 2.18i·11-s − 2.09·12-s − 3.76i·13-s − 3.14·14-s − 0.0557·15-s − 0.542·16-s − 4.22·17-s + ⋯
L(s)  = 1  − 1.59i·2-s + 0.389·3-s − 1.55·4-s − 0.0369·5-s − 0.622i·6-s − 0.526i·7-s + 0.888i·8-s − 0.848·9-s + 0.0591i·10-s − 0.658i·11-s − 0.605·12-s − 1.04i·13-s − 0.841·14-s − 0.0143·15-s − 0.135·16-s − 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $-0.995 - 0.0957i$
Analytic conductor: \(2.78677\)
Root analytic conductor: \(1.66936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 349,\ (\ :1/2),\ -0.995 - 0.0957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0563226 + 1.17329i\)
\(L(\frac12)\) \(\approx\) \(0.0563226 + 1.17329i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (-18.5 - 1.78i)T \)
good2 \( 1 + 2.26iT - 2T^{2} \)
3 \( 1 - 0.674T + 3T^{2} \)
5 \( 1 + 0.0827T + 5T^{2} \)
7 \( 1 + 1.39iT - 7T^{2} \)
11 \( 1 + 2.18iT - 11T^{2} \)
13 \( 1 + 3.76iT - 13T^{2} \)
17 \( 1 + 4.22T + 17T^{2} \)
19 \( 1 - 3.48T + 19T^{2} \)
23 \( 1 - 7.19T + 23T^{2} \)
29 \( 1 - 5.38T + 29T^{2} \)
31 \( 1 - 2.72T + 31T^{2} \)
37 \( 1 - 1.29T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + 6.57iT - 43T^{2} \)
47 \( 1 - 9.84iT - 47T^{2} \)
53 \( 1 + 0.274iT - 53T^{2} \)
59 \( 1 - 5.46iT - 59T^{2} \)
61 \( 1 + 1.99iT - 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 0.386iT - 71T^{2} \)
73 \( 1 - 1.57T + 73T^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 - 0.123T + 83T^{2} \)
89 \( 1 + 5.22iT - 89T^{2} \)
97 \( 1 + 11.3iT - 97T^{2} \)
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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

−11.03271639562891496737943655765, −10.39654130088375948709087163232, −9.303453849824071080342751414178, −8.613191080486152274969150049126, −7.47793666649986844730544285216, −5.90906970890145083203406029815, −4.56221542409915201316327141598, −3.32980467003088659062907391954, −2.62625734744705411427363325860, −0.793255392034683864738167690804, 2.52333657711852834836597726588, 4.32489936590081275232547002713, 5.31631499241246233396515936553, 6.34474994135301750151838478625, 7.15364258780600492177951720690, 8.134148350844283938632076406985, 8.979905731662486168325103313136, 9.488224137667588829556879808662, 11.16070325788820277862017620659, 11.97647398203545071973296856184

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