Properties

Label 2-350-175.73-c1-0-18
Degree $2$
Conductor $350$
Sign $0.873 + 0.485i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
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  + (0.933 − 0.358i)2-s + (3.06 + 0.160i)3-s + (0.743 − 0.669i)4-s + (−0.514 − 2.17i)5-s + (2.92 − 0.949i)6-s + (−2.31 + 1.27i)7-s + (0.453 − 0.891i)8-s + (6.40 + 0.673i)9-s + (−1.26 − 1.84i)10-s + (0.555 + 5.28i)11-s + (2.38 − 1.93i)12-s + (−5.88 + 0.931i)13-s + (−1.70 + 2.02i)14-s + (−1.22 − 6.75i)15-s + (0.104 − 0.994i)16-s + (−1.52 − 2.34i)17-s + ⋯
L(s)  = 1  + (0.660 − 0.253i)2-s + (1.77 + 0.0928i)3-s + (0.371 − 0.334i)4-s + (−0.230 − 0.973i)5-s + (1.19 − 0.387i)6-s + (−0.875 + 0.482i)7-s + (0.160 − 0.315i)8-s + (2.13 + 0.224i)9-s + (−0.398 − 0.584i)10-s + (0.167 + 1.59i)11-s + (0.689 − 0.558i)12-s + (−1.63 + 0.258i)13-s + (−0.455 + 0.540i)14-s + (−0.317 − 1.74i)15-s + (0.0261 − 0.248i)16-s + (−0.369 − 0.568i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.873 + 0.485i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.873 + 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.76228 - 0.716317i\)
\(L(\frac12)\) \(\approx\) \(2.76228 - 0.716317i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.933 + 0.358i)T \)
5 \( 1 + (0.514 + 2.17i)T \)
7 \( 1 + (2.31 - 1.27i)T \)
good3 \( 1 + (-3.06 - 0.160i)T + (2.98 + 0.313i)T^{2} \)
11 \( 1 + (-0.555 - 5.28i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (5.88 - 0.931i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (1.52 + 2.34i)T + (-6.91 + 15.5i)T^{2} \)
19 \( 1 + (-3.21 + 3.56i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (1.27 + 3.32i)T + (-17.0 + 15.3i)T^{2} \)
29 \( 1 + (-1.77 - 0.575i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.759 - 3.57i)T + (-28.3 - 12.6i)T^{2} \)
37 \( 1 + (1.64 + 2.03i)T + (-7.69 + 36.1i)T^{2} \)
41 \( 1 + (2.55 - 3.52i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-3.78 - 3.78i)T + 43iT^{2} \)
47 \( 1 + (-5.08 - 3.30i)T + (19.1 + 42.9i)T^{2} \)
53 \( 1 + (0.135 - 2.58i)T + (-52.7 - 5.54i)T^{2} \)
59 \( 1 + (0.586 - 0.261i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (-4.14 + 9.30i)T + (-40.8 - 45.3i)T^{2} \)
67 \( 1 + (-3.62 + 2.35i)T + (27.2 - 61.2i)T^{2} \)
71 \( 1 + (1.58 - 4.88i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.95 + 4.81i)T + (15.1 + 71.4i)T^{2} \)
79 \( 1 + (-0.241 - 1.13i)T + (-72.1 + 32.1i)T^{2} \)
83 \( 1 + (8.17 + 4.16i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (5.84 + 2.60i)T + (59.5 + 66.1i)T^{2} \)
97 \( 1 + (-5.25 + 2.67i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.92896785688099876230690698303, −9.996902314527444559980080755242, −9.513849075014683137144926478540, −8.895916548993889808116886542365, −7.56401117399271983109482283129, −6.93096549734141106723510562385, −4.94336753407651459230297892123, −4.32894151765472717552768517026, −2.93236792309158495841437793947, −2.09403316084175863726668225971, 2.47817844828194534663736206656, 3.33957927840012422706049487501, 3.92407944478601677120135447992, 5.86617499664115245435696871240, 7.06905523800267644769169932219, 7.61439481337940845361542799801, 8.571575387070422471317494960039, 9.741468323182035409434011908408, 10.40993915890410241897737032708, 11.75283042285689294087215189960

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