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Properties

Label 2-354-177.176-c5-0-99
Degree $2$
Conductor $354$
Sign $-0.329 - 0.944i$
Analytic cond. $56.7758$
Root an. cond. $7.53497$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + (1.84 − 15.4i)3-s + 16·4-s − 87.9i·5-s + (7.38 − 61.9i)6-s − 193.·7-s + 64·8-s + (−236. − 57.1i)9-s − 351. i·10-s + 206.·11-s + (29.5 − 247. i)12-s + 754. i·13-s − 774.·14-s + (−1.36e3 − 162. i)15-s + 256·16-s − 1.40e3i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.118 − 0.992i)3-s + 0.5·4-s − 1.57i·5-s + (0.0837 − 0.702i)6-s − 1.49·7-s + 0.353·8-s + (−0.971 − 0.235i)9-s − 1.11i·10-s + 0.514·11-s + (0.0591 − 0.496i)12-s + 1.23i·13-s − 1.05·14-s + (−1.56 − 0.186i)15-s + 0.250·16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.329 - 0.944i$
Analytic conductor: \(56.7758\)
Root analytic conductor: \(7.53497\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :5/2),\ -0.329 - 0.944i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7806806583\)
\(L(\frac12)\) \(\approx\) \(0.7806806583\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4T \)
3 \( 1 + (-1.84 + 15.4i)T \)
59 \( 1 + (-2.40e4 + 1.17e4i)T \)
good5 \( 1 + 87.9iT - 3.12e3T^{2} \)
7 \( 1 + 193.T + 1.68e4T^{2} \)
11 \( 1 - 206.T + 1.61e5T^{2} \)
13 \( 1 - 754. iT - 3.71e5T^{2} \)
17 \( 1 + 1.40e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.39e3T + 2.47e6T^{2} \)
23 \( 1 - 859.T + 6.43e6T^{2} \)
29 \( 1 + 2.01e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.20e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.37e4iT - 6.93e7T^{2} \)
41 \( 1 - 2.92e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.46e4iT - 1.47e8T^{2} \)
47 \( 1 + 8.21e3T + 2.29e8T^{2} \)
53 \( 1 + 2.66e4iT - 4.18e8T^{2} \)
61 \( 1 - 3.93e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.29e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.91e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.75e4iT - 2.07e9T^{2} \)
79 \( 1 - 5.71e4T + 3.07e9T^{2} \)
83 \( 1 + 7.75e4T + 3.93e9T^{2} \)
89 \( 1 - 1.35e4T + 5.58e9T^{2} \)
97 \( 1 - 3.42e4iT - 8.58e9T^{2} \)
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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

−9.656521893529774222685255846552, −9.117537570764241175995950671612, −8.074186014401660072528374802864, −6.75579667238704153047948692883, −6.29221470118477233993929480893, −5.02586292584181082858695411086, −3.94898501604475971308407398853, −2.58984620631719925562222157532, −1.30144617832745738109603440410, −0.14734864418827776334100305703, 2.53204188380307222314573093491, 3.36575749126275695996965304018, 3.89109545847579855491372385295, 5.60109062752653733312110568691, 6.33239412650471949238857536269, 7.15770080756233955358413604063, 8.600174189827263607150299115367, 9.796984107468174021270017428696, 10.66773911613205622936427952206, 10.77338204587229029834827421814

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