Properties

Label 2-175-175.108-c1-0-0
Degree $2$
Conductor $175$
Sign $-0.277 - 0.960i$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
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  + (0.0938 + 1.79i)2-s + (−2.61 − 2.11i)3-s + (−1.21 + 0.127i)4-s + (1.96 + 1.06i)5-s + (3.54 − 4.88i)6-s + (−1.70 + 2.02i)7-s + (0.219 + 1.38i)8-s + (1.72 + 8.13i)9-s + (−1.71 + 3.62i)10-s + (2.72 + 0.579i)11-s + (3.43 + 2.23i)12-s + (0.649 + 1.27i)13-s + (−3.78 − 2.85i)14-s + (−2.89 − 6.94i)15-s + (−4.84 + 1.02i)16-s + (1.22 + 0.470i)17-s + ⋯
L(s)  = 1  + (0.0663 + 1.26i)2-s + (−1.50 − 1.22i)3-s + (−0.605 + 0.0636i)4-s + (0.879 + 0.475i)5-s + (1.44 − 1.99i)6-s + (−0.643 + 0.765i)7-s + (0.0775 + 0.489i)8-s + (0.576 + 2.71i)9-s + (−0.543 + 1.14i)10-s + (0.821 + 0.174i)11-s + (0.992 + 0.644i)12-s + (0.180 + 0.353i)13-s + (−1.01 − 0.763i)14-s + (−0.747 − 1.79i)15-s + (−1.21 + 0.257i)16-s + (0.297 + 0.114i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ -0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503023 + 0.668803i\)
\(L(\frac12)\) \(\approx\) \(0.503023 + 0.668803i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.96 - 1.06i)T \)
7 \( 1 + (1.70 - 2.02i)T \)
good2 \( 1 + (-0.0938 - 1.79i)T + (-1.98 + 0.209i)T^{2} \)
3 \( 1 + (2.61 + 2.11i)T + (0.623 + 2.93i)T^{2} \)
11 \( 1 + (-2.72 - 0.579i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.649 - 1.27i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-1.22 - 0.470i)T + (12.6 + 11.3i)T^{2} \)
19 \( 1 + (-0.229 + 2.18i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (1.79 - 0.0940i)T + (22.8 - 2.40i)T^{2} \)
29 \( 1 + (1.22 + 1.68i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.74 - 3.92i)T + (-20.7 + 23.0i)T^{2} \)
37 \( 1 + (-0.0698 + 0.107i)T + (-15.0 - 33.8i)T^{2} \)
41 \( 1 + (6.08 + 1.97i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.62 - 2.62i)T + 43iT^{2} \)
47 \( 1 + (2.63 + 6.85i)T + (-34.9 + 31.4i)T^{2} \)
53 \( 1 + (-7.79 + 9.62i)T + (-11.0 - 51.8i)T^{2} \)
59 \( 1 + (-0.660 - 0.733i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-5.97 - 5.37i)T + (6.37 + 60.6i)T^{2} \)
67 \( 1 + (1.09 - 2.86i)T + (-49.7 - 44.8i)T^{2} \)
71 \( 1 + (-4.78 + 3.47i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.03 - 0.670i)T + (29.6 - 66.6i)T^{2} \)
79 \( 1 + (-3.47 + 7.80i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (6.21 - 0.983i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (-2.11 + 2.35i)T + (-9.30 - 88.5i)T^{2} \)
97 \( 1 + (-13.4 - 2.12i)T + (92.2 + 29.9i)T^{2} \)
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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

−13.14452607599504347462728458761, −12.03604748398632492909989036968, −11.32900971061006243139439302120, −10.06665287979182304721285912978, −8.617066661793233178382058508898, −7.15648719095670677464113880848, −6.60277862735116365257526998580, −5.95445378534834451950638830313, −5.15984459977698012903905113082, −2.02826508278341169478955474470, 0.971893047643615735618022109360, 3.52947587407153190165107113627, 4.44931498164182704102770907502, 5.78348000641277455279608154756, 6.66903383128037092845891393129, 9.229297016086815082624806379728, 9.906274812435908826104823901651, 10.42885285488520522941763556090, 11.30794093394763684604124797580, 12.18140626930253938131332349760

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