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Properties

Label 2-425-85.9-c1-0-21
Degree $2$
Conductor $425$
Sign $0.998 - 0.0580i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
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  + (1.52 + 1.52i)2-s + (1.06 − 2.56i)3-s + 2.67i·4-s + (5.54 − 2.29i)6-s + (2.90 − 1.20i)7-s + (−1.03 + 1.03i)8-s + (−3.32 − 3.32i)9-s + (−4.70 + 1.94i)11-s + (6.86 + 2.84i)12-s − 1.39·13-s + (6.27 + 2.60i)14-s + 2.18·16-s + (−3.66 + 1.88i)17-s − 10.1i·18-s + (3.63 − 3.63i)19-s + ⋯
L(s)  = 1  + (1.08 + 1.08i)2-s + (0.613 − 1.48i)3-s + 1.33i·4-s + (2.26 − 0.937i)6-s + (1.09 − 0.454i)7-s + (−0.366 + 0.366i)8-s + (−1.10 − 1.10i)9-s + (−1.41 + 0.587i)11-s + (1.98 + 0.820i)12-s − 0.386·13-s + (1.67 + 0.695i)14-s + 0.545·16-s + (−0.889 + 0.457i)17-s − 2.39i·18-s + (0.833 − 0.833i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.998 - 0.0580i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.998 - 0.0580i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.93106 + 0.0851357i\)
\(L(\frac12)\) \(\approx\) \(2.93106 + 0.0851357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (3.66 - 1.88i)T \)
good2 \( 1 + (-1.52 - 1.52i)T + 2iT^{2} \)
3 \( 1 + (-1.06 + 2.56i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-2.90 + 1.20i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (4.70 - 1.94i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 1.39T + 13T^{2} \)
19 \( 1 + (-3.63 + 3.63i)T - 19iT^{2} \)
23 \( 1 + (-2.37 - 5.73i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (2.65 - 6.41i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-2.33 - 0.966i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (2.97 - 7.17i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.83 + 9.25i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.08 + 2.08i)T - 43iT^{2} \)
47 \( 1 + 5.08T + 47T^{2} \)
53 \( 1 + (-2.93 - 2.93i)T + 53iT^{2} \)
59 \( 1 + (-0.594 - 0.594i)T + 59iT^{2} \)
61 \( 1 + (2.29 + 5.54i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 4.10iT - 67T^{2} \)
71 \( 1 + (9.66 + 4.00i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.549 + 0.227i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (6.77 - 2.80i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-3.35 - 3.35i)T + 83iT^{2} \)
89 \( 1 + 14.1iT - 89T^{2} \)
97 \( 1 + (2.37 + 0.982i)T + (68.5 + 68.5i)T^{2} \)
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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

−11.58297315804623984492573832365, −10.42732901795550144479587706103, −8.798214147826636980707018131778, −7.83348246676900537224457647458, −7.37376635580880098577545973668, −6.80011475580759904185847694009, −5.39418867390337593825747873606, −4.73146973460941805081110325767, −3.10553151816771956318169531614, −1.69782657724477080780570071137, 2.31915403294888610889806885635, 3.06147479027464410988638022275, 4.30047456639345690805668188048, 4.92918776277227285157196776256, 5.63729454564738204626583805120, 7.86037527672366510683178817268, 8.587910765713399498707379037352, 9.766551186856703866056978059371, 10.47937863415949122348785235459, 11.16749858108613137564702551224

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